(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Equations for the calculation of chromatographic figures of merit"

EQUATIONS FOR THE CALCULATION OF 
CHROMATOGRAPHIC FIGURES OF MERIT 



By 

JOE PRESTON FOLEY 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOI 
OF THE UNIVERSITY OF FLORIDA IN 
PARTIAL FULFILLMENT OF THE REQUIREMENTS 
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
1983 



To Mom and Dad, for their love, support, and encouragement; 
and to my sister Barbara, for leading the way. 



ACKNOWLEDGMENTS 

First and foremost, I would like to thank my research director, 
John G. Dorsey, not only for helping me with various research projects, 
but for being a special friend— for always finding the time to listen 
and for always doing the "extra things." 

Second, I want to express my gratitude to Thomas J. Buckley and 
Sharon G. Lias of the National Bureau of Standards for their help and 
the use of their facilities in preparing this document. 

Finally, I want to thank all the friends I made in Gainesville for 
making my graduate education at the University of Florida the happiest 
and most satisfying time of my life. 



111 



TABLE OF CONTENTS 

CHAPTER PAGE 

ACKNOWLEDGMENTS iii 

ABSTRACT v i 

1 INTRODUCTION 1 

Overview „ 1 

Chromatographic Peak Characterization 1 

Limit of Detection 8 

2 GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) 

FUNCTION AND RELATED DATA 9 

Introduction 9 

EMG Evaluation 9 

Background 9 

Evaluation of the Integral Term 10 

Obtaining Universal EMG Data 16 

Background 16 

Measurement of the Pertinent Peak Parameters . „ 17 

Comparison of Universal EMG Data 22 

Conclusion „ 24 

3 EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC 

FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS 25 

Introduction 25 

Derivations 25 

Experimental , 28 

Apparatus , 28 

Procedure 28 

Results 30 

Recommended CFOM Equations , 30 

Other CFOM Equations 33 

Discussion _ 39 

Detailed Discussion of Precision 39 

Why Measure At 10$ Peak Height? 41 

General Aspects , 46 

Conclusion , 55 



IV 



4 CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY 56 

Introduction . 5b 

Identifying Current Problems 56 

Literature Survey Results 56 

Numerical Example 58 

Solving the Problems „ 60 

Eliminating Mistaken Identities 60 

Choosing a Model 63 

Using the Correct Units 66 

Converting to Chromatographic Reference Conditions 70 

The Numerical Example - Revisited 79 

Conclusion 81 

5 SUGGESTIONS FOR FUTURE WORK 82 



APPENDICES 

A DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMG-U 84 

B UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, and 0.50 87 

C DERIVATION OF V inj>max 95 

REFERENCES 96 

BIOGRAPHICAL SKETCH 100 



Abstract of Dissertation Presented to the Graduate School 
of the University of Florida in Partial Fulfillment of the 
Requirements for the Degree of Doctor of Philosophy 



EQUATIONS FOR THE CALCULATION OF 
CHROMATOGRAPHIC FIGURES OF MERIT 



By 

JOE PRESTON FOLEY 
December 1 983 



Chairman: Dr. John G. Dorsey 
Major Department: Chemistry 



The measurement and interpretation of several chromatographic 
concepts and parameters, hereafter referred to as chromatographic 
figures of merit (CFOMs) , are improved via equations and concepts 
developed in this work. 

The previous uses of the exponentially modified Gaussian (EMG) 
model in chromatography are briefly reviewed. A method for evaluating 
the EMG function and a set of algorithms for obtaining universal data 
are presented and are shown to be simpler and easier to use than those 
previously reported. The corresponding BASIC program, EMG-U, is also 
briefly discussed. 

By use of the exponentially modified Gaussian (EMG) as the skewed 
peak model, empirical equations based solely on the graphically 
measurable retention time, t fi , peak width at 10% peak height, W Q ,, and 

vi 



the empirical asymmetry factor, B/A, have been developed for the 

accurate and precise calculation of CFOMs characterizing both ideal 

(Gaussian) and skewed peaks. These CFOMs include the observed 

efficiency (number of theoretical plates), N ; the maximum efficiency 

sys 

attainable if all asymmetry is eliminated, N„ v ; the EMG peak 
parameters, t^, a-,, and t; the first through fourth statistical moments; 
the peak skew and peak excess, Yg and Y„; and two new CFOMs — the 
relative system efficiency, RSE, and the relative plate loss, RPL. 
Equations for the number of theoretical plates and the variance (second 
central moment) are accurate to within ±1.5% for 1.00 <_ B/A £ 2.76. 
width and B/A at 10% peak height are recommended. 

The current problems with the LOD concept in chromatography are 
reviewed. They include confusing the LOD with other concepts in trace 
analysis; the use of arbitrary, unjustified models; the use of 
concentration units instead of units of amount; and the failure to 
account for differences in chromatographic conditions (bandwidths) when 
comparing LODs. 

Two models are proposed for calculating the chromatographic LOD. A 
new concept, the standardized chromatographic LOD, is introduced to 
account for differences in chromatographic bandwidths of experimentally 
measured LODs. The standardized chromatographic LOD is shown to be a 
more reliable CFOM than the conventional (non-standardized) 
chromatographic LOD. 



Vll 



CHAPTER 1 
INTRODUCTION 



Overview 

Chromatography is a well-known method for the separation and 
quantitation of chemical moieties from a (sample) mixture. Over the 
years several concepts and parameters, hereafter referred to as 
chromatographic figures of merit (CFOMs), have been introduced to 
characterize the separation and quantitation. Unfortunately, some of 
the CFOMs are often difficult to estimate [those which characterize 
chromatographic peaks]; others are ambiguous [e.g., the limit of 
detection (LOD)]. The goal of the present work, which is introduced in 
more detail in the following two sections, is the improvement of the 
measurement and interpretation of these chromatographic figures of 
merit. 

It is beyond the scope of this work to introduce or review the 
development of these CFOMs from either a historical or theoretical point 
of view. Such discussions and references to additional discussions may 
be found elsewhere (1-10). 

Chromatographic Peak Characterization 

In recent years there has been considerable interest in the 

characterization of experimental chromatographic peaks. Presented in 

Table 1.1 are the names, symbols, and general expressions that have 

evolved for the parameters used in chromatographic peak 



p 

•r-I 

CD 

s 

O 

01 

CD 
Cm 
3 
bO 
•H 
Eb 

O 
•H 

£ 
Q. 
cd 
Cm 
bO 
O 

■p 
cfl 
S 

o 
s^ 
XI 
o 

c 
o 
a 

s 
o 
o 



c 

o 

■H 

m 

CO 
0) 
SL, 

a 

<d 



cd 

Cm 
CD 

a 

CD 
h0 



CD 
O 

a 

(0 
•H 

Cm 

CO 
> 

CNJ 

CD 

B 
•H 

P 

C 

o 

•H 

P 

c 

CD 
-P 
CD 
Cm 



o 

X 

e 

GO 



a 

03 



a -h 

3 o 

a t, 

•H P 

X Ctl 

cti a> cu 

aoa 

M X X 
cd cd cd 

C CD CD 
D, Q. O. 



« «- s 

■P S P 



.Q 








cd 








H 


Cm 


CO 






CD 


CD 






-P 


P 






CD 


cO 






S 


rH 


a 




cd 


a 


E 




Cm 




•H 




CO 


.-) 


P 




ft 


cd 








o 


c 






•H 


C 






4>> 


1-J 






CD 


P 






Cm 


c 






O 


a- 






a) 


P 






.a 


a> 






jj 


S- 



a 
o 

•H 

p 
a 
cd 

Cm 

Cm 

P 

XS 

b0 
•H 
(D 

x 
a 

CO 
CD 



CO 
P 

C 
•H 

O 

a 

a 

o 

■r-I 
P 

o 

(D 
H 
Cm 

c 



a 
o 

Cm 
Cm 

G 

s 

cd 

Cm 



CD 


CD 




TD 




CO 


p 


Cm 


Cm 


CO 






CD 


TJ 


3 


=3 


CD 


CO 




•H 




bO 


b£> 


P 


P 




-P 


,~v 


■H 


•H 


O 


C 




•H 


J-> 


[XI 


Ex. 


a 


CD 




C 8 


— * 






CD 


bO 




XI 


CD 


CD 


■o 


C 




CO V 




CD 


CD 




CO 




3 




CO 


CO 


cd 


P 




cr 
■o 

CD 

p 
cd 
H 
CD 
Cm 




< 








p 


•a 




\ 


a, 


cd 


X) 


■— *• 


c 


o 


DQ 


X! 


13 

p 

X 
bO 
•H 


3= 


x 


cd 

CO 

p 

a 

CD 

% 

6 

cd 
o 

■r-i 


s 


Cm 




CD 






P 




O 




X 






CO 




P 










■H 




O 




X3 






P 




cd 




CD 






cd 




Cm 




•r-I 

Cm 






p 

co 




>t 




•H 




a 






Cm 




O 


CD 


o 




*— > 


P 




CD 


CO 


■H 




cd 


CD 




a 


cd 


■p 




CD 


1 




CO 


X 


o 
C 




Cm 
cd 


>} 




p 


p 


3 






CO 




CO 


cd 


<M 




id 


cd 


P 










cd 




X! 


X! 


-C 


CD 




CD 


rH 


bO 


P 


P 


a, 




a 


cd 


•H 


-o 


13 


cd 




> » 


a 


CD 


•H 


■H 


XI 






•H 


X! 


» 


3 


CO 




XI 


Cm 












-p 


■H 


^ 


it 


JsS 


ZA 




o 


D. 


cd 


cd 


cd 


CO 




Cm 


a 


CD 


CD 


CD 


CD 




CD 


CD 


a 


D. 


a. 


Q. 




N 



o 
s 

V. 

4-5 



o 

2 
■\ 
p 

X3 



P 

X 
CM 



O 

S 

*■>» 
p 



-p 

* — <■ 

X 

a 




CM 

\ 

CO 
C\J 

2 

s. 
on 

2 



i-n 
f 
c\j 

CM 



o 

Cm 

-P 
G 
CD 
O 

a 
cd 

CD 

a. 



p 

W 
Cm 
•H 

Cm 



CM 

S 




O 

C 

cd 

•H 
Cm 

cd 
> 



P 

a 

CD 

s 
o 

s 



cd 

Cm 

p 
C 

CD 

o 

G 
CM 



>- 



CO 



P 

a 

CD 

a 
o 
s 

rH 

cd 

Cm 

p 

C 
CD 
Q 

X 
P 

G 



3 
CD 

CO 

i, 
cd 
CD 

a 



CO 
CO 
CD 
O 
X 
CD 

id 

cd 

CD 

a 



characterization. The graphical chromatographic parameters are 
illustrated in Figure 1.1. 

These CFOMs have been estimated either manually using graphical 
measurements made directly from the chromatogram or by a computer 
following data acquisition. Both methods have advantages and 
disadvantages. 

Manual methods were used exclusively at first and are employed 
quite extensively today. For arbitrary peak shapes, they are accurate 
for only five CFOMs: t fi , B/A, h p , W b , and W . If a Gaussian peak shape 
is assumed, however, then M^ = t R , and Mp is only a function of W,, W , 
or Mq and N may subsequently be calculated. Except for higher 
even central moments, the remaining CFOMs are zero for Gaussian peaks. 

For real chromatographic peaks, it is almost always a mistake to 
assume a Gaussian peak shape. Experimentally these ideal, symmetric 
peaks are rarely, if ever, observed due to various intracolumn and 
extracolumn sources of asymmetry (5,11-23). Kirkland et al. have shown 
that the plate count can be overestimated by as much or more than 100% 
if any of the three most common Gaussian-based equations are 
employed (23). 

Computer estimation methods are more accurate than common manual 
methods for a given CFOM but are not available to every chromatographer. 
The general approach taken has been one of peak statistical moment 
analysis (6,11,22-24). Via relatively simple algorithms all the CFOMs 
may be determined quite accurately, though the precision of the second 
and higher central moments is seriously affected by baseline noise (25). 

The failure of the Gaussian function as a peak shape model for real 
chromatographic peaks led to the search for a more accurate model and 



H 

LJ 



< 

LJ 
CL 



Q 
LU 
N 



O 




Figure 1.1. Graphical chromatographic parameters shown at peak height 
fraction a = 0.10. Except for A, B, and B/A at a = 0.10, 
all width related measurements are subscripted with the 
value of a to prevent ambiguity. 



the eventual acceptance of the exponentially modified Gaussian (£MG), a 
function obtained via the convolution of a Gaussian function and an 
exponential decay function which provides an asymmetric peak profile. 
The development, characterization, and theoretical and experimental 
justification of this model have been thoroughly reviewed (21,22,26,27). 
Previous chromatographic studies (11,12,14,15,17-23,25-31) involving the 
EMG function, summarized in Table 1.2, demonstrate the utility of this 
skewed peak model. 

Adoption of the EMG peak-shape model has improved the estimation of 
the CFOMs. A new algorithm for the computer-based peak moment analysis 
has been derived (25) and tested (22) which is less sensitive to 
baseline noise and the uncertainty of peak start/stop assignments. More 
recently, Barber and Carr described a manual method for CFOM 
quantitation which requires the graphically measurable retention time 
tpi peak width W, empirical asymmetry factor B/A, and successive 
interpolations from three large-scale universal calibration curves 
(31,32). 

The primary objective of this part of the present study is the 
development, using the EMG model, of accurate equations for CFOM 
calculation dependent solely on tp, Wq « , and B/A. The need for 
computerized data acquisition is thus circumvented, and, in addition, 
CFOM calculation via these equations is expected to be faster and more 
precise than the other accurate manual method since no graphical 
interpolation is required. 

The previously reported methods for evaluating the EMG function 
(14,16,19,27) and obtaining chromatographic peak data (19,26,31) were 
too inaccurate or too unwieldy to use in the present study, which 



rH 


0) 


































CD 


O 


*t 


in 


*- 


t-- 


so 


ca 


o 


CO 


CTi 


o 


T- 


t*~ 


LO 


CM 


'vO 


ro 


O 
2 


c 

CO 

3 














CM 


CM 


CM 


en 


CM 


CM 


CM 


CM 


CM 


CM 




































0) 


































•• V 


Ch 


































a 


0) 


































25 


as 


































53 




































— * 


























CO 

34t 










c 


























cfl 










cd 



































CO 


■H 




















bO 






a 








o 


co 












m 








3 














rH 


co 












3 








•H 






o 








cd 


3 












•H 








43 






•H 








o 


CO 












P 




CD 




43 






43 










o 




c 








43 

•H 




a 

cO 










a 

cfl 


bO 






43 

c 


-o 




c 








Cm 




43 










3 


3 






3 


0) 
















CO 











to 


•H 






O 


•r-J 




c 








CD 








> 






o 


3 






o 


Cm 




•H 








> 


CD 


-is! 




3 


CO 




43 











•H 










(D 


L 


a 


CO 




3 







CO 


■n 




CO 





T3 




>> 






a 


3 


cd 


CD 




O 


> 




a 


CO 




T3 


43 


o 




P 






cO 


O 


43 


a 






3 




o 


o 




O 


CO 


s 




•H 






43 




CO 






3 


3 




3 


3 




43 


rH 






> 






CO 


CO 




o 




O 


O 




43 


43 




43 


a 


>> 




•H 








CD 


X 


■H 




CO 






a 












rH 




-P 






JM 


3 


cd 


43 




43 


bO 






T3 


a 


a 


3 


H 




•H 






CO 


CO 


CD 


a 




a 


3 




■3 


3 


CO 




o 


cd 




n 






CD 


3 


a 


CO 




cfl 


•H 







cfl 


43 


43 




•i-i 




e 






a 


CT 




3 




cc 


a 




a 


jO 


CO 


3 


s 


P 




<u 








CO 


o 


bO 




1 


a 




a 












a 




CO 




bO 


a 




■H 


o 


CO 


3 


cfl 




cfl 


CJ 


X 


B 


M 


a) 


■o 






C 


•H 


P 


43 


43 


a 


o 


rH 




r-i 


•H 


CO 





CO 


a 


CD 


3 




•H 


43 


CO 


a 


cti 


CO 


43 


3 




3 


43 





u 




o 


>1 


c 


CJ 


3 


a 


CO 


cO 


a 


3 


3 










a 


a 


3 


X 


a 


o 


i—: 


J 


CD 


cd 


CD 


3 


o 


bO 


CD 


> 




> 


cfl 




CO 


cd 


X 


,— 1 


<i-: 


o 


-o 


U 


rH 


bO 


u 


O 


is 


o 


O 


O 


3 


3 


cfl 





w 


a 






CO 


bo 




o 


a 


43 






C3 




bO 


o 





a 




S 


•c 


3 


o 


O 


CO 


43 


o 


cd 


cfl 


». 




i>> 


O 




a 







w 


3 


•r-4 


£ 


43 


•H 


CO 




s 


•H 


"O 


3 


rH 


43 







Cm 


43 




cd 




X) 


cO 


> 


a 


3 


o 


> 





■H 


to 


CO 


43 


3 


o 


P 


rH 




>> 




a 




o 


o 


3 




p 




3 


a 


cd 


o 









CD 


CJ 


-o 


o 


CO 


3 




43 


CO 


3 


>> 


O 


o 


3 


•H 


43 


bO 


n 


a 


c 


a 


3 


id 


43 


0) 


O 


id 


O 


O 


3 


3 




43 


o 


C 


o 


•H 


<D 


cfl 


43 


CO 


O 


a 




cfl 


43 


3 


43 


-3 


3 


cd 





•H 


s 


p 


■H 


JQ 


O 


CD 




■H 


Cm 


CD 


CO 





CO 


O 


O 


3 


Cm 


CO 






a 






a 


3 


43 


O 


a 


•H 


•H 






rH 


cd 


Cm 


a 


CJ 


CD 


•H 


O 


3 




O 








-o 


O 


Cm 


Cm 


Cm 


a 







s 


M 


Cm 


43 


O 


u 




CD 


3 


o 




•H 


O 


o 









CO 


« 


3 


Cm 






a 


CO 


CO 


O 


CD 


>» 


Cm 






XS 


CO 


• ~ 







o 


CD 


CO 


CD 




CD 


3 


•H 




rH 


Cm 


3 


3 


3 




o 


•H 


x: 


a 




e 


a 


■a 


S 


O 


43 


■o 


rH 





O 


O 


cd 


M 


J 


T3 


o 


n 


3 


o 


3 





•H 


a 


cO 


CD 


CO 




•H 


•H 









3 


■H 


CD 


5 


■H 


rH 


a 


-P 


CO 


a 


> 


P 


3 


P 


43 


CD 


o 


3 


-P 


A 


3 


3 


43 


o 


a 




CD 


•H 


rH 


3 


O 


cfl 


CO 


S 


•H 


■H 


co 


3^ 




rH 


3 


> 


cd 


3 


3 


X 










bO 


N 


3 


43 








3 


O 


_o 




rH 


O 




o 


CO 


a 


CO 


■H 


■H 


rH 


c 


CO 


o 


c 


o 


o 


•H 


3 


Eh 


•H 


3 


3 


(D 


3 


P 


P 


3 


o 





43 


•H 


■H 


-p 




3 


o 


CD 


43 


CD 


a 


3 


3 


O 


CO 





> 


43 


o 


JS 




o 


Cm 


-P 


43 


> 


CO 


T3 


a 


3 


43 








43 










Q. 


43 


CD 


O 


C 


o 


o 


X 


3 


cd 


3 


CO 


Cm 


> 


o 


-a 


3 


Cm 


cfl 


O 


-P 




O 







cO 


O 






3 


Cm 


3 


CO 


cfl 




Cm 


3 


CD 


CD 


a 


o 


43 


Cm 


rH 


O 


CO 


Cm 


•H 





■H 


3 


CD 


Cm 


CD 


hfl 


•i-> 


13 


o 




CD 


o 


CD 


CD 


CD 


O 








cd 


T3 


O 




o 


43 




•H 


c 


T3 




Eh 


3 


3 




Cm 





rH 


3 






C 


P 


3 


Cm 


-p 


£: 




3 






cfl 


3 


O 


a 


Cfl 


o 


Cm 


a 


a 


CO 


CO 


O 


CO 


3 


Cm 


O 


Cm 


Cm 


3 


O 




3 


O 




O 


o 


3 


a 






a 


rH 


O 


•H 


O 


O 


a* 


•H 


CO 


rH 


•H 


T3 




CO 


rH 


o 




CO 


■H 


o 




43 






00 


P 


■H 


o 


43 





CO 


■H 


O 


u 




-P 


S 


o 


43 


3 


43 


43 




3 


CO 


> 





> 


43 


3 


o 


-C 




o 


3 


CO 


o 


rH 


o 


o 


43 


H 


>^ 




3 


O 


o 


cd 


cd 


cj> 




CD 


CD 


3 


CD 


o 


CD 


CD 


00 


O 


rH 


73 


o 


3 





a 


3 






Q-i 


P 


43 


Cm 


CO 


Cm 


: M 


cfl 


CO 


CO 


CO 





a 


Cm 


e 


43 


• 




Cm 


CD 


K 


Cm 


CD 


Cm 


Cm 


CD 





3 





x: 


B 


Cm 


o 


X 


CM 

* — 




CD 


X5 


CD 


CD 


3 


CD 


CD 


rH 


3 


cd 


■o 


p 


•H 


CD 


o 


CD 




rH 


t, 


ff\ 


CTi 


v£> 


CO 


CA 


OS 


CFl 


O 


o 




CM 


CM 


C- 


[». 


f— 


r— 


43 


cO 


in 


in 


■J3 


-D 


VO 


MD 


vO 


fr- 


t- 


c— 


n— 




t- 


C— 


c— 


c— 


CO 
H 





ca 


o> 


CTi 


CA 


CA 


ca 


a\ 


<Ti 


Oi 


CA 


CA 


o 


a\ 


a% 


CA 


•CA 



C\J *- 



O 



■H 

-P 



CD 

-P 

Q, 
CO 

O 







^! 








cri 








2 








b 


-p 






■H 


•H 






jC 


Eh 






a 


CD 






Ct) 


S 






^ 








hC 


Cm 






o 


O 






-p 






*■ — » 


cO 


CO 




3 


2 


CD 




CD 


o 


S-. 




•H 


u 


3 




> 


SZ 


M 




CD 


O 


•H 




Sh 




«H 




w 


a 








s 


O 




g 


SI 


•H 




, o 




_c 




■H 


Cm 


Q. 




-P 


o 


CO 




!U 




m 




O 


C 


bo 




JP 


o 


o 




w 


•H 


-p 




•H 


-P 


en 




-a 


D 


a 






H 


o 




-■4 


O 


u 




cfl 


> 


,a 




CD 


c 


o 




Q, 


o 








o 


t, 


• 


.H 


CD 


o 


T3 


aj 


■o 


Cm 


0) 


-P 






-a 


C 


M 


02 


c 


CD 


CO 


C 


CD 


B 


O 


O 


-p 


D 


•rH 


•H 


X 


S-. 


-C 


■P 


CD 


-P 


a 


CO 


1 


CO 


c0 


3 


1 


C 


L, 


CT 


CM 


•H 


bO 


■CD 


<~ 








CD 








.H 


CO 


i — 


cn 


.O 


t— 


03 


CO 


cd 


CT\ 


CT\ 


cy> 



8 

employed an Apple II Plus microcomputer. This necessitated, therefore, 
the development of a simpler method for evaluating the EMG function and 
the incorporation of a simpler, more accurate, and more general set of 
algorithms for obtaining the EMG data of interest. 

Limit of Detection 

The limit of detection (LOD) is generally defined as the smallest 
concentration or amount of analyte that can be detected with reasonable 
certainty for a given analytical procedure. Though arguably the most 
important figure of merit in trace analysis, the LOD remains an 
ambiguous quantity in the field of chromatography. Detection limits 
differing by orders of magnitude are frequently reported for very 
similar (sometimes identical!) chromatographic systems. Such huge 
discrepancies raise serious questions about the validity of the LOD 
concept in chromatography. 

The primary objective of this part of the present work is to 
restore the integrity of the LOD concept, to make the chromatographic 
LOD a reliable, meaningful figure of merit. This will be accomplished 
in two steps: First, the major sources of the discrepancies in 
chromatographic detection limits, i.e., the current problems with the 
LOD concept, will be identified. Second, each problem will be addressed 
and eliminated (or circumvented). 



CHAPTER 2 
GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION 

AND RELATED DATA 



Introduction 
This chapter describes the improvements achieved by this study in 
evaluating the EMG function and in obtaining the EMG data of interest. 
The results of the present work are compared with those previously 
obtained. In addition, the corresponding BASIC computer program, EMG-U, 
is listed and discussed briefly in Appendix A. Universal EMG data are 
tabulated in Appendix B. It is hoped that this will facilitate the use 
of the EMG function, whenever applicable, in modeling studies in 
chromatography or any other area. 

EMG Evaluation 
Background 

Description of the EMG function . It is beyond our scope to derive 
the EMG function from first principles. Those who so desire should see 
the treatments given by Sternberg (11, pp. 250-253) or Kissinger et al. 
(12, pp. 159-162). Their results are shown below in eq 2.1. 



z/(2) 1/2 
*EMG XW = *-i\UQ{£) --/ij expLu.muQ/ T ;- _ ^ -o — -c G , v ' 



h £MG (t) = [Atf G (2) 1/2 A] exp[0.5(a r /T) 2 - (t-t r _)A] /exp(-x 2 )dx (2.1) 



/• 



PE 1 E 1 I 1 

where z = (t - t G )/a Q - On/x . (2.2) 

Equation 2.1 shows that the EMG function is defined by three parameters: 



10 

the retention time, t Q , and standard deviation, a Q , of the parent 
Gaussian function; and the time constant, t ? f the exponential decay 
function. In addition, the quotient t/Oq is a fundamental measure of an 
EMG peak's asymmetry. The arbitrary constant A determines the amplitude 
of the function. Note that the right hand side of eq 2.1 can be broken 
into 3 parts: the pre-exponential term (PE.,), the exponential term 
(£.]), and the integral term (I,). 

Discrepancies. A literature survey which I conducted revealed 
three discrepancies in eq 2.1. In the first instance, a factor of 2 
difference observed in the pre-exponential term, PE,, is relatively 
unimportant because this affects only the zeroth statistical moment 
(peak area). All other parameters, including higher order moments which 
are normalized by the zeroth moment, are unchanged by this factor of 2. 
The second and third discrepancies are serious, however. Both were 
observed in the denominator of the second quotient, (t - t Q )/T, within 
the exponential term, E.,. In one case, an additional factor of 2 was 
present; in the other, a Q was added to this denominator and omitted from 
the numerator of the pre-exponential term, p£,. These errors invalidate 
those expressions for the EMG function. 

Evaluation of the Integral Term 

Range of z. The methods used to evaluate eq 2.1 have frequently 
been omitted from the EMG literature. Since the evaluation of the first 
two terms is straightforward, the reported methods differ only in the 
manner in which the integral, Ij, is determined; the I-j approximations, 
in turn, depend on the value of z in the upper limit of I.. It is 
convenient to group the range of possible z values into three regions: 
a) z <. -3; b)-3 <. z < 4; and c) z >. 4 . In region c, the definite 



11 



integral equals the constant (tt) 1/2 to within 0.01? and is thus 
virtually independent of z. In regions a and b, however, the relative 
value of the definite integral is highly dependent on z. Numerous 
techniques permit the accurate evaluation of L in region b. In region 
a, however, as z becomes more negative, it becomes more difficult to 
approximate the integral to the same (high) relative accuracy. Thus it 
is pertinent to examine the practical minimum values of z for EMG peaks. 

As seen from eq 2.2, z depends on a) the normalized difference 
between the time of interest, t, and t Q ; and b) the reciprocal of the 
peak asymmetry, (T/a G )" 1 . For a given peak shape (constant t/o g ), z 
will therefore be smallest (most negative) at the starting threshold of 
the peak, which may be conveniently defined as that time (t) on the 
leading edge of the EMG peak where the value of the function Ch Pun (t)] 
is a specified fraction, B, of its maximum value, i.e., h EMG (t)/h p = B. 
Moreover, the minimum z value will decrease as the starting threshold is 
decreased. This is shown in Table 2.1 where minimum z values are 
tabulated for EMG peaks with asymmetries ranging from 0.1 to 3. Another 
trend illustrated is that for a given starting threshold, z ■ increases 
with increasing asymmetry (Va G ). 

Previous methods. Given this wide range of z values, how is I 1 
(in eq 2.1) evaluated? Except for a vague reference to an unspecified 
polynomial approximation (30), all previous methods for calculating I 1 
employ different techniques for different values of z. For moderate z 
values (e.g., regions b and c, above), some methods utilized the well- 
known identity 



/. 



exp(-y 2 )dy = 0.5U) 1/2 [1 + erf(x)] (2.3) 



12 



Table 2.1. Minimum z Values Needed to Evaluate EMG Peaks for 
Various Asymmetries (r/a Q ) and Starting Thresholds (B) a 



0.001 0.01 0.1 



T/0 G 



0.10 b -12.9 -12.1 

0.15 -10.3 - 9.6 - 8.7 

0.20 - 8.6 - 7.9 - 7.0 

0.25 - 7.5 - 6.9 - 6.0 

0.30 - 6.8 - 6.2 - 5.3 

1.00 - 4.4 - 3.6 - 2.7 

3.00 - 3.6 - 2.8 - 1.9 



a See eq 2.2 and text for description of z, T/a Q , and B. 
Values for z have been rounded. 

b An underflow error occurs z < -13, thus preventing its 
measurement. 



13 



(where erf is the error function) in order to take advantage of error 
function subroutines resident in the computers used (19,27). Others 
used eq 2.3, but approximated the error function by interpolation from a 
set of tabulated areas (14,18). For small z values (region a, above), 
the error function techniques were sufficiently inaccurate to warrant 
the use of other methods instead. In nearly every instance some type of 
asymptotic series was employed (14,19,27). In the lone exception, a 
Gaussian function was substituted for the EMG function for any part of 
the peak profile where very small z values were encountered (18). 

This work. The approach taken in this study is to transform the 
integral in eq 2.1 via change of variable [x = y/(2) 1/2 ] to 



(tt) 1/2 /exp(-y 2 )dy/(27r) 1/2 



The EMG function can now be written as 

h EMG (t) = tAa G (27T) 1/2 / T ][E 1 (see eq 2.1)] / exp(-y 2 )dy/(2Tr) 1/2 (2.4) 

■ -/-co 

?E 4 * h " 

The integral in eq 2.4 can be approximated by a polynomial approximation 
l 4 (z<0) = NF(z) P(q) and l 4 (z>.0) = 1- l 4 (z<0), where 
NF(z) = exp(-z 2 /2)/(27r) 1/2 , P(q) = ^q + b 2 q 2 + b 3 q 3 + b^q 4 + b 5 q 5 , 

•1 

q = (1 + pz)~ , and p,b 1 ,...,b 5 are constants given in Table 2.2 (33). 

Comparison. The values obtained for 1^ are compared to the true 
values (34,35) in Table 2.3 from z = -10 to z > 3.9. In addition, since 

In = ^/(tt) 172 (2.5) 

they can be compared to values obtained for I* via an asymptotic 



14 



Table 2.2. Constants in the Polynomial Approximation 
for lu in eq 2.4 a 



p = 0.2316419 t> 3 = 1.781477937 

b 1 = 0.319381530 b 4 = -1.821255978 

b 2 = -0.356563782 b 5 = 1.330274429 



a See text immediately after eq 2.4 or reference 33. 



15 



Cd 






















t 










=T 






















>l 










m 
CM 



































o 










en 


r — 


* — 


.. — 


* — 


> 




T3 






a 


Cd 


<r— 


o 


O 


=r 


o 


o 


O 


O 


o 


•H 




G 






o 


PC 




















.p 




cd 






■H 




ro 


C\J 


^ — 


o 


o 


o, 


O, 


a, 


o 


o 










-13 


M 












+1 


+1 


+1 


+1 


CD 




„ 






cd 




+ 


+ 


+ 


+ 


+ 










a 




CM 






3 














V 


V 


V 


V 


W 




1 






O" 






















CD 










M 






















g 




A| 






C 


A 




















_. 




N 






•H 


G 




















H 




G 






J* 


o 
























o 






H 


3 


CO 

on 


CM 


•JO 


o 


i 














Cm 






G 


W 


i 


1 


1 


I 


o 










cd 




0) 






O 


3 


o 


O 


o 


o 


•r— 














3 






Cm 


o 

•rl 


*■■ 


*"" 


K— 


,— 


X 


1 


f 


! 


1 


-3- 




+3 






C 


> 


X 


X 


Xi 


X 




! 


I 


i 


! 


h- 1 










O 


CD 










vO 




1 


1 


1 






„ 






•H 


G 


CO 


CM 


CM 


c— 


■X) 










^ 




^r 






cd 
S 


Q. 


t— 


VO 


CM 
vO 


CO 

CTv 


oo 










G 


CM 


H 

O 


! 




■H 


•=r 




















O 


■N. 


4J 






X 


H 




















Cm 


^~ 




v| 




O 
























^-v 


XJ 






G 






















CO 


(s 


CD 


N 




& 






















C 




Sh 






D. 






















o 


\ 


cd 


G 




"* 


£4 






















M 


g 


O 
Cm 




rH 


r^ 


pr 


^r 


vO 


o 


in 










to 




o 






rd 


o 


m 


CM 


i — 


1— 


l 










CD 


1] 


o 


^ 




•H 


3 


8 


! 


I 


1 


o 










G 






G 




S 




en 


O 


o 


o 


T— 










Q, 


St 


^ 


O 




o 


CQ 


^~ 


i — 


1 — 


•c— 












X 


I—! 


u 


3 




c 


•H 










X 










<D 




o 






>» 


X! 


X 


X 


X 


X 




c— 


O 


<M 


o 




J-J 


3 


CO 




rH 


4J 










CXi 


CM 


o 


e— 


o 


G 


cd 




3 




o 




oo 


t- 


CO 


o 


cO 


CM 


o 


c— 


o 


O 


,G 


w 


O 




Q-. 


JSP 


OO 


t— 


CM 


C3-. 


•"" 


O 


in 


cr. 


o 


Cm 


4J 


•H 

j3 


■H 
> 




<D 


H 


V— 


D— 


VO 


en 


oo 


o 


o 


O 


*— 


T— 


CD 


-U 





• 


-G 






















• 


-P 




G 


o 


-P 






















CM 


O 

c 


^T 


a 


i 


Cm 






















■o 




H 


M 




O 


* — V 




















c 
cd 


■a 

a 


G 


C\J 


V 


>i 


^r 




^r 


VO 


o 








**% 


, — - 




cd 


■H 




N 


O 


m 




CM 


t — 


< — 


in 






m 


in 


a. 






l ^~ K 




cd 


— ' 




I 


I 


1 


i 






on 


en 


^r 


c— 


M 


[= 


G 


G 






o 


o 


o 


o 






■ — * 


v_^ 


■ 


CM 


o 




O 


3 


d) 




* 


1 — 


* — 


^— 










CM 




G 


V, 


Q— r 


o 


3 












c— 


o 


CM 


o 







G 






a 


G 


x> 


X 


X 


X 


X 


CM 


o 


fc— 


o 


c 


a 


0) 


M 


-a 


< 


4-> 












CM 


o 


e'- 


o 


OJ 


G 






<D 








vD 


CM 


CTi 


CM 


CO 


in 


er* 


o 


• 


CD 





O 


4J> 


• 


* 




















CM 


S-, 


bO 


JJ 


G 


oo 


■=r 




c— 


vO 


en 


ro 


O 


o 


o 


<— 




CD 


cd 




O 


• 


H 




















CO 


Cm 


4-5 


■a 


O, 


C\J 






















cr 
CD 


CD 


c 





G 


CD 

G 





























o 


ca 




r— 1 




















cr. 


CD 


CD 


G 


Q, 


4-5 


rj 




CM 


O 


x> 


vO 


.=}- 


CM 


o 


CM 


• 





CD 


0) 


S 


o 


cd 


N 


T ~" 


■— 














oo 


CO 


CO 


eu 


o 




H 




] 





! 


1 


1 









Al 


cd 


X5 


o 


a 


xT 



16 

series (27), the most accurate method reported for z < -3. (Comparison 
with values for I 1 obtained via error function techniques when z > -3 
(Ijj > 10"°) is not illustrative since the maximum absolute error in the 
polynomial approximation Iu(z) is estimated to be + 7.5 x 10~ 8 .) 

Table 2.3 shows that our method for evaluating 1^ is exceptionally 
accurate for moderate values of z. Moreover, it compares favorably with 
the asymptotic series method for evaluating I,, except for z < -8 where 
the latter method is somewhat better. Re-examination of Table 2.1 
shows, however, that for all practical purposes z > -8 whenever T/a G J> 
0.2. Thus our method for evaluation of 1^ can be used to evaluate the 
EMG function to within 1% or less for i/o n > 0.2. 

While being slightly less accurate than the most accurate previous 
method, the new technique for evaluating the EMG function is much more 
convenient than any of the previous methods. Only one simple subroutine 
requiring just a few programming lines (see lines 2990-3160 in Appendix 
A) is needed, whereas the other methods require at least two 
subroutines, if implemented on any computer without a built-in error 
function routine (i.e., nearly all microcomputers and many 
minicomputers) . 

Obtaining Universal EMG Data 
Background 

Using the polynomial approximation for the integral in eq 2.4, the 
EMG function can be evaluated over the entire practical time range. 
Depending on the data required for the modeling process of interest, the 
EMG peaks could be generated "on the fly" as needed. Alternatively, the 
necessary EMG data could be generated (and stored) in advance and 
accessed when needed. Though the storage requirements may seem 



17 



prohibitive for the latter, given a T/a Q value and a peak height 
fraction, a, three quantities completely specify an EMG peak (31). 
Figure t.1 shows an EMG peak with its pertinent graphical parameters. 
Regardless of the retention time, t Q , and standard deviation, o of the 
(unconvoluted) parent Gaussian peak (not snown) , (B/A) . W_/a P . and (t D - 
^G^ a G are universal constants so long as T /° G and a remain fixed. 
Recent work has utilized these universal data sets almost exclusively 
(3D. 

Experimentally, three parameters must be determined in order to 
calculate the universal data: t R , t«, and t B (see Figure 1.1). Note 
that t R must be obtained before t. or t g because h p = h EMG (t R ) is needed 
for the latter. 

Measurement of the Pertinent Peak Parameters 

Previous methods . In the past, t R nas been determined by one or 
more of the following methods: a) peak displacement data and knowledge 
of t G (19); b) differentiation of h £MG (t) and solving for roots (26); 
and c) least squares fitting of the top of the peak with a quadratic 
gram polynomial (26,31). 

Once t R has been found, t A and tg can be located. In the only 
method reported previously, two points [t.,, hg^tt^], [t 2 , h EMG (t 2 )] 
are found so that t 1 < t A < t g (or t 1 < to < to). Linear interpolation 
yields the approximation for t« (or t B ). 

This work . The approach for finding these quantities is based on 
iterative search mechanisms, as the flowchart in Figure 2.1 shows. In 
the case of t R , initial time limits are easily found using the fact that 
t R is always greater than t Q . The EMG function is then evaluated from 



18 



Find initial time limits 




Y 



J±. 



Calculate new time limits 




N 



t = t + dt (t R , t B only) 
t=t-dt (t A only) 



Y 



-> 



Calculate value of t (and h ) 



f A° r V 



N 



_^L 



Decrease value of dt 



Figure 2.1. Simplified flowchart for locating t (and h ), t , or t . 
The El'IG peak parameters must be input before beginning this 
search. 



19 

the lower time limit to the upper time limit in increments of dt. When 
the maximum is found, new lower and upper time limits [given by 
t( current) - 2dt and t(current), respectively] closer to the peak 
maximum are set, the time increment is decreased, and the search is 
begun again. The retention time, t R , is approximated by t(last) - 

dt . . 
mm 

The algorithm for estimating t A (or tg) is similar to the t. (or 
t B ) search algorithm previously discussed in that the time limits are 
analogous. Since t A < t R < tg, initial values for t 1 and to are easily 
determined. To locate t A (or tg) , the EMG function is then evaluated 
from t s tg to t^ in decrements of dt (or from t 1 to to in increments of 
dt) until h EMQ (t) < ahp. New values for t^ and to [given by t( current) 
and t(current) + dt, respectively] closer to t A (or tp) are then set, 
the time decrement (increment) is decreased, and the search is repeated. 
This is continued until t< and t~ are known to the desired precision; t» 
(or tg) is then given by (t 1 + tg)/2. 

The maximum error in the values of t R , t A , tg and related universal 
EMG quantities obtained via these search algorithms is presented in 
Table 2.4. In all cases the error is dependent on the smallest (most 
precise) time increment (or decrement), dt ^ i used in the last 
iteration of each search. In theory, dt min could be as small as 
desired. Due to the finite precision of computers, however, the time 
increment, dt , would ultimately be reduced to such a low value that 

h EMG (t ) ~ h EMG (t+dt). (2.6) 

Henceforth the algorithms would cease to function accurately, if at all. 



20 



Table 2.4. Maximum Errors in the Universal 
EMG Data and Selected Component Parameters 

Parameter 3 Maximum Error 



H ± dt min 



*!• fc B ± 1/2 dt min 

A = t R -t A , B = t B -t R ± 3/2 dt m . n 



V°G = (V t A> /a G ± dt min /a G 



( W /ff G ± dt min /G G 



a A, B, to » t., and tg defined in Figure 1.1; 

a,,, t P , and dt .„ described in text. 
G' U' mm 



21 

As dt is decreased, the t R search algorithm will fail first, since 
the slope of the EMG function is smallest in the region of t Q . Somewhat 
smaller (more precise) dt min 's could be employed in the searches for t. 
and tg before the algorithm breakdown described by eq 2„6 would occur. 
The increase in precision of t A and t fi is probably not worth the effort, 
however, since two out of the three universal EMG data expressions are 
dependent on the least precise quantity, t R . 

The minimum usable value of dt depends on the precision of the 
computer employed and on the value of o Q chosen. In this study, the 
experimentally measured minimum ratio of dt/ov, was 0.0002 for 0.1 < t/o q 
< 3 using single precision arithmetic. Multiple precision capabilities 
would allow a still lower dt/cr„ ratio to be used. 

Comparison . The algorithms for t R , tj,, and tg may be compared as 
follows : 

1. With the exception of the quadratic least squares method for finding 
tg, all of the methods for obtaining t R , t,, and tg are designed for 
simulated data (essentially no noise). 

2. The two algorithms for calculating t. (or tg) are quite similar. 
Both require two points which closely bracket the desired peak height 
fraction, a, and both are relatively unbiased. The subsequent 
interpolation performed in the previously described algorithm is 
potentially more precise than the averaging of the final time limits in 
the proposed search algorithm. If t» and tg are already known as 
precisely as or more precisely than t R , however, additional improvements 
in their precision, even if realized, will not yield significant 
increases in the precision of B/A and (t R -t G )/ a G . 

3. Our approach for determining t R , though crude, is superior to the 
other three methods previously discussed for the following reasons: 

i) It is a general algorithm. Whereas methods a and b are specific to 

the EMG model, our search mechanisms will work for that and other peak 

models as well. 

ii) It is accurate and unbiased. In contrast, the quadratic least 

squares fitting method, though general, suffers from a small, but 

nevertheless observable bias (due to a determinate error) which 

increases with increasing peak asymmetry (26). 

iii) It is easy to understand and implement. The other methods are 

unnecessarily complex, though they admittedly have the potential 

for greater precision. 



22 



4. The proposed search algorithms for determining t R) t A , and to are 
superior to the previous methods because they can be de-bugged more 
easily. By having the time and the value of the EHG function printed 
every time the EMG function is evaluated, the programmer can literally 
watch the computer perform the search. Since the search logic is so 
simple, programming errors are easily detected. Upon elimination of the 
errors, the print statement may be removed. 

Comparison of Universal EMG Data 

Table 2.5 shows representative sets of universal EMG data obtained 
from this study and from a previous work (32) which utilized the 
quadratic least squares method and the interpolation method for the 
location of t R and t A (or t g ), respectively. The precision is reported 
for our data (in terms of maximum errors) and is assumed to be no worse 
than + 1 in the least significant digit of the previously reported data. 
Several points should be noted: 

1 . Although this difference is slight or non-existent at high 
asymmetries, the previously obtained universal data are somewhat more 
precise. This is expected since the algorithms used in locating t R , t., 
and t B are potentially more precise than those developed here. 

2. The data sets are in excellent agreement for all three universal EMG 
quantities at low asymmetries (t/ct„ <. 0.5). This agreement is 
especially significant at i/o Q =0.1, because it shows that the moderate 
errors introduced by the polynomial approximation for K in eq 2.4 when 
-12 < z < -» (see Table 2.3) are not transmitted to the universal EMG 
data. 

3. Whereas the W a /a Q data reported previously are consistent with the 
corresponding data of this study over the entire range of T /cr p , the 
remaining data sets are discordant for x/o Q 2 1.0. Relative to the 
current data sets, the previous ones for (t R -t G )/a Q and (B/A) appear to 
be slightly overestimated and underestimated, respectively. This 
discrepancy is due to the use of a least squares fitting method in the 
previous measurement of t R which overestimates this quantity for EMG 
peaks (26) and other types of skewed peaks. This bias increases from an 
insignificant value at low x/a Q to an observable one at r/a _> 1.0, 

4. Despite the differences noted above, the general interlaboratory 
agreement is quite good. Though the current data are more accurate, 
either set of EMG data can be used with confidence for modeling studies. 



23 









CD 


















! 




o 


T- 


<*— 


tO 


«r— 


to 








1 




c 


o 


o 


o 


i— 


X— 








1 




CD 


o 


o 


O 


o 


o 








I 




Eh 


o 


o 


o 


o 


o 








I 




a) 


• 


• 


• 


* 


• 








•i 




Cm 


o 


o 


o 


o 


o 








i 




Cm 


3 


+ 


+ 


+ 


+ 








1 




•H 


















1 
1 




Q 


CM 
















a i 


co 




o 


c- 


o 










co 


C i 


3 




^— ' 


cn 


to 


■L 


o 






-P 


"s. I 


O 


Jrf 


cn 


(^n 


en 


I — 


o 






£ 


'-* 1 


•H 


Eh 


ro 


CO 


e- 


a^ 


c— 






O 1 


> 


o 


en 


CM 


cn 


i — 


l — 








•P 1 


<P 


SB 


o 


=T 


vO 


o 


<M 






- — V 


1 1 


S- 




• 


m 


• 


• 


■ 






O 


CC 1 


Oh 




o 


o 


o 


T— 


v— 






'£> 


-p 1 




















w o 


^-^ 1 




















-^^ H— 


I 




















■ 


1 




O 


o 


CM 


^r 


o 


-T 






c o 


1 




>> 


St 


CO 


fr» 


CO 


in 






cC 


I 


CO 


T3 


ON 


CM 


cn 


1 — 


^ — 






•H II 


I 


•rH 


3 


o 


^r 


X) 


o 


C\J 






« 





X! 


-p 


« 


• 


• 


II 


• 






CO (0 


1 


H 


CO 


o 


o 


o 


T — 


^ — 






3 






















CO -P 






















O CO 






















•a *-> 






CD 






^m 


,— 






B 


CD CM 






O 


CD 


<D 


o 


o 


CD 




>. 


•i-i on 






c 


a 


C 


o 


o 


C 




T3 


4-1 -^ 






CD 


o 


o 


o 


o 


O 




3 


•H 






Jh 


a 


c 


« 


• 


C 




4-> 


■a >> 













o 


o 






CO 


O T3 






Cm 






+ 


+ 








S 3 






Cm 














m 


-P 






•H 














■H 


>,CO 






a 














X! 


rH 




CO 




cn 


C\J 


o 


^r 


CM 




jp 


.H CO 




3 




t— 


C\J 


m 


Ln 


O 






CO 3 




o 


-^ 


St 


C— 


on 


in 


VO 




Cm 


•H O 




■H 


Eh 


•r— 


oo 


to 


^ — 


CM 




O 


-P -H 


*p ■ 


> 


O 


00 


c— 


tO 


ien 








C > 


G I 


CD 


3 


« 


« 


• 


• 


O 




CO 


CD CD 


•v. 1 


Eh 




=T 


St 


in 


c— 


T 


m 


•P 


C Eh 


3 1 


a, 














>t 


nJ 


o a. 


















T3 


X) 


a 
















C\J 


3 




X CO 




X2 


<M 


CM 


CM 


^r 


o 


-P 


a 


w 






>> 


=r 


0- 


on 


m 


to 


CO 


o 


x) 




CO 


•a 


^ — 


CO 


to 


i — 


CM 




\ 


rH C 




•H 


3 


ro 


c— 


tO 


en 


• 


co 


^— s 


CO CO 




x! 


.p 


* 


* 


• 


• 


o 


•H 


o 


CO 




H 


co 


^T 


^r 


LTl 


C- 


-. — 


X! 


-p 


Eh >. 


















-P 


1 


a? td 




















cn 


> 3 


















<M 


•p 


•H .p 






CD 












o 




C CO 


E 




O 


C\J 


^~ 


LTl 


^=r 


CM 






Z3 


!: 




a 


o 


o 


O 


^~ 


C\l 


s 


m 


CO 


;j 




CD 


o 


o 


O 


o 


o 


3 


a 


Cm -H 


:i 




Eh 


o 


o 


o 


o 


o 


■P 


o 


O X! 


I 




<D 


• 


• 


• 


• 


■ 


CO 


V, 


-P 


] 




Cm 


o 


o 


o 


o 


o 


T3 


s 


a 


I 




Cm 


+ 


1 


I 


! 


1 






o e 


| 




•H 












<S 


H 


CO o 


1 




Q 


x> 










•s. 


H 


•H Eh 


1 






en 


a 








CQ 


CO 


U <w 


B 


CO 




< — 


1 — 


to 










CO 


I 


3 




c— 


a 


en 


^=r 


r— 


X! 


Eh 


a -o 


i 


O 


Js! 


■^r 


JD 


in 


^ — 


ir- 


o 


O 


a cp 


D 


■H 


Eh 


t— 


CM 


* — 


^r 


on 


co 


Cm 


o c 


: 


> 


o 


o 


Cn 


to 


!X^ 


VO 


CD 




O -H 


l 


CD 


3 


o 


O 


in 


o 


c— 




CM 


cd 


1 


Eh 




« 


• 


■ 


• 


• 


Eh 


o 


■ -p 


< 1 


Oh 




T— 


i — 


* — 


CM 


CM 


O 


O 


LO X) 


V. | 
















Cm 


o 


• O 


03 1 




















CM 


J 
















T3 


o 




! 






CO 


CO 


on 


=r 


=r 


CD 




CD 


1 






o 


o 


o 


o 


o 


-P 


+1 


rH 


« 






o 


o 


o 


o 


o 


Eh 




X2 


S 






o 


o 


o 


o 


o 


O 


CD 


CO 


I 






• 


• 


• 


• 


• 


a 


M 


H 


1 


CO 


o 


o 


o 


o 


o 


CD 


cfl 




1 




>> 












Eh 






I 


CO 


-a 


+1 


+1 


+1 


+1 


+1 




CO 




1 


-H 


3 












Eh 


Eh 




! 


X! 


-p 


on 


C— 


H— 


in 


cn 


o 


O 




E 


H 


co 


* — 


OJ 


CM 


LTl 


in 


Eh 


M 




1 






o 


en 


UD 


cn 


•o 


Eh 


M 




I 






o 


o 


CO 


o 


o- 


CD 


CD 




1 






• 


* 


• 


* 


• 








1 








* — 


^~ 


CM 


CM 


B 
3 
S 
■H 


S 
3 
S 
•H 




G° 






•r— 


LTt 


o 


o 


o 


s 


a 




"v 






» 


* 


• 


• 


• 


s 


2! 




H 






o 


o 


■*— 


CM 


m 


CO 


X! 



24 

Conclusion 
Though the exponentially modified Gaussian (EMG) model has already 
been employed in numerous studies in chromatography as Table 1.2 shows, 
its usage might have been still more extensive had it not been for some 
confusing discrepancies and for the overly complex methods used for its 
evaluation reported previously. Hopefully the clarification of these 
discrepancies and presentation of a simple method for evaluating the EMG 
function and obtaining universal data will encourage more scientists in 
all fields to use the EMG model, when appropriate. 






a 



CHAPTER 3 
EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC 
FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS 



Introduction 
Adoption of the EMG model for chromatographic peak characterization 
results in a new set of chromatographic figures of merit (CFOMs) which 
re listed in Table 3.1. These CFOMs consist of fundamental and 
derived EMG parameters, the latter containing explicit expressions for 
the first through fourth statistical moments defined previously in 
Table 1.1. Included among these new CFOMs are the following: the 
retention time, t Q , and standard deviation, a Q , of the associated parent 
Gaussian peak from which the skewed peak is derived; the exponential 
modifier, t; the fundamental ratio, t/Oq, which characterizes peak 
asymmetry; the observed efficiency (number of theoretical plates) of a 
given (asymmetric) chromatographic system, N ; the maximum efficiency 
a given system could achieve, N max , if all sources of asymmetry were 
eliminated; and finally, two CFOMs proposed originally in this work 
which demonstrate how peak asymmetry drastically reduces chromatographic 
efficiency— the relative system efficiency, RSE, and the relative plate 
loss, RPL. 

Derivations 

If the estimate G obtained from using a Gaussian peak shape 

equation is used to approximate the true value T of a CFOM for an 

asymmetric peak, the relative error RE which results is defined as 

25 



26 



Table 3.1. Chromatographic Figures of Merit Based on 
the Exponentially Modified Gaussian (EMG) Model 

Fundamental 

t , o Qf t, x/a G 

Derived 



N sys = H 2 '(% 2 + t2 ) 



N max ~~ (V a G )£ 



RSE = ^ T sys /N max) <W 2 = ° G 2/M 2 

RPL = C(N max - N sys )/N max ] 'W* = 1 " RSE = t2/M 2 



M| = t Q + x 



M, = r 2 + t 2 



M 3 = 2 T 3 



M 4 = 3ct q 4 + 6a Q 2 t 2 + 9 t 4 



Y s = M 3 /M 2 3/2 



f E = M 4 /M 2 2 - 3 



27 



RE = (G - T)/T (3.1) 
which can be rearranged to give 

T = G/(fiE + 1) (3.2) 
Thus, the true value T and the Gaussian approximation G for the CFOM are 

related by the correction term (RE + 1) in the denominator of eq 3.2. 
Kirkland et al. have shown that 

RE = f(T/a Q ) (3.3) 

for the three popular Gaussian-based methods for determining plate 
counts of a system (23). Equation 3-3 should hold, in fact, for any 
CFOM for which a Gaussian approximation exists except t Q . Since Barber 
and Carr have shown (31) that 

t/a G = f(8/A) (3.4) 

successive substitution of eqs 3-3 and 3.4 into eq 3.2 yields 

T = G/[f(B/A) + 1] (3.5) 

Although the exact form of f(B/A) is unknown, a least squares curve 

fitting of an RE versus B/A plot can give an excellent approximation. 

The above approach was used for calculation of N , a rt and M . 

sy s u c. 

Following this, t was calculated (see sixth equation, Table 3.1) by 

x . [M 2 - a Q 2]V2 (3<6) 

For the determination of t Q , the universal relationship 

(t R - t G )/ff Q = f(T/a Q ) (3.7) 

previously reported (31) was rearranged and combined with eq 3.4 to give 

fc G = H ~ a G f(B/A) (3.8) 



28 



where f(B/A) was approximated by a least squares fit of (t R - t Q )/a G 
vs. B/A. 

Since t R , B/A, and W 0>1 are graphically measurable, all the 
remaining CFOMs can be calculated once the fundamental parameters o„, t, 
and tn have been determined. 

Experimental 
Apparatus 

An Apple II Plus 48K RAM microcomputer was programmed in BASIC for 
EMG peak generation. A curve-fitting program available from Interactive 
Microware (P.O. Box 771, State College, Pa., 16801) with linear, 
geometric, exponential, and polynomial capabilities was used for the 
unweighted least squares fitting of various data sets. 

Procedure 

EMG Peak Generation. Except as otherwise noted, values of A = 1, 
t G = 100, and ° G = 5 were used in eq 2.4 for EMG peak generation. The 
x /°q ratio was varied from 0.1 to 3 in 0.05 increments producing data 
equivalent to 59 peaks. The times for t R , t A , and tg (see Figure 1.1) 
at a = 0.1, 0.3, and 0.5 for each of the 59 peaks were determined to 
within 0.001, using the simple search algorithm described in Chapter 2; 
corresponding values of W and B/A were then computed. 

Development of the CFQM equations . Textfiles of fiE(N , M , a r ) 
vs. (B/A) 0<1)0>3j0>5 , (t R - t G )/° G vs. (B/A) 0#1>0#3)0-5 , and RSE vs. 

^ B/A ^0.1 ,0.3,0.5 were made for °- 1 i- T/cr G ^- °'3- The Gaussian relations 
CT G = w 0. i/' 4 - 2 9 1932 and M 2 = (W Q# ^4 .291932) 2 were used in the Gaussian 
approximations for ^ sy3 , a G , and M 2 at a = 0.1. Similar equations were 
used for M 2 an d a Q at a - .3, 0.5. The true values of N gys , a Q , M 2 , 



29 



and RSE were computed from the known values of x, o«, t Q , and t R for a 
given peak. 

The CFOM equations were developed by unweighted least squares curve 
fitting. The relationship between several quantities [e.g., RE(N), 
(B/A) a , to a / CT Q] and t/o q has been shown previously (23,31) to be 
nonlinear for < V° G < 0.5 and nearly linear for 0.5 <. T /°n <. 3. 
Because similar relationships were observed in some of the textfiles 
above, least squares fitting was limited to about the same t/cu range 
0.5-3 (1.09 < B/A <_ 2.76) except for the RSE vs. B/A textfile where 
the complete set of values was used in the regression analysis. 
Although the least squares fittings of the various B/A textfiles were 
initially judged by visual inspection and the coefficient of 
determination (square root of the correlation coefficient), their final 
evaluation was based on the accuracy and simplicity of the resulting 
CFOM equations. 

For ease of use, the CFOM equations were first simplified 
algebraically and then by the successive rounding of numerical 
coefficients. The occasional nearness of the decimal coefficients to 
whole numbers was exploited. For example, if f(B/A) = 1.02(B/A) + 0.69, 
then for 1 .09 <. B/A i 2.76, the much simpler function 
f(B/A) = B/A + 0.72 is approximately the same (exactly if B/A = 1.5) and 
the accuracy of this function is not significantly affected. 

All CFOM equations were simplified as much as possible — any more 
rounding of the coefficients will result in appreciably greater error. 

Evaluation of the CFOM equations . The accuracy of the CFOM 
equations was evaluated in terms of four parameters, listed in 
decreasing order of importance: the percent relative error limits 



30 



($RELs) which represent the maximum possible error of the CFO'M equations 
within the specified B/A range; the mean percent relative error or bias, 
J&RE = E%RE/n; the average magnitude of the percent relative error, 
Z|$RE|/n; and the standard deviation of the percent relative error. 

The precision of the empirical, EMG-based CFOM equations and three 
Gaussian CFOM equations was calculated via error propagation theory and 
is reported as percent relative standard deviation (S&RSD). The 
required precision estimates of t R , W a , and (B/A) (the graphically 
measurable quantities) were obtained using data from a previous 
study (32). 

Results 
Recommended CFOM Equations 

Listed in Table 3.2 are the empirical CFOM equations based on t R , 
W 0.1' and B/ ^ A measurements (see Figure 1.1) which we recommend. 

Accuracy. Using equation 1 in Table 3.2, the true efficiency of a 
chromatographic system, N sys , may be estimated to within + 1.5$ for both 
Gaussian and exponentially tailed peaks within the asymmetry range 1.00 
1 B/A i. 2.76. Equations for Mg, t Q , and M 1 are equally if not more 
accurate over this asymmetry range. 

All CFOMs except RPL, Mg, Y g> and Y E can be estimated to within 
± 5% for 1.09 <. B/A <. 2.76, the asymmetry range over which most of the 
curve fitting was performed. 

All CFOM equations are accurate to within + 5% for 1.19 <. B/A <_ 
2.76, and 18 out of 21 are accurate to within ±2%. 



31 



n 


uo 


CM 


^r 


o 


a -p 


■ 


• 


• 


« 


CO -H 


CM 


CM 


CM 


CM 


K S 










•H 


+1 


+1 


+1 


+1 


a* rH 











u 



u 



Cm 



CM 


o 


in 


* 


• 


■ 


O 


CM 


St 


+1 


-M 


+1 



Cw 



m 


in 






jQ 


o 


o 




o 











-p 


o • 


Lfl 


IT\ 


LO 


LA 


o 


in 


o 




o 




LH 


■H 


• T~" ' 


• 


• 


• 


• 


• 


• 


• 




• 




• 


B 


*- 1 


o 


i — 


o 


PO 


LO 


^r 


<o 




4 — 




*— 


■H -H 


+ 


+ 


+ 


+ 


+ 


+ 


+ 


+ 




+ 




-:- 


0) r-i 


•* 
















-o 








M 


-o 


•» 


*• 


~ 


~ 


•» 


M 


n 


LO 


•» 


O 


■«, 


M 


LTv • 


LA 


O 


o 


o 


in 


o 


LO 




lh 




o 


■*«. O 


■ o 


• 


• 


• 


• 


• 


■ 


• 


o 




CM 




f-, 


T— *— 


<— 


* — 


t — 


r— 


o 


T~ 


o 




o 




y3 


0) 


1 1 


E 


1 


] 


;i 


I 


I 


1 


+1 


! 


+1 


f 



cd 

-P 

■H 

t, 

0) cd 



U CQ 

.p ^ 

CD 



S 

E 



11} 

to 



CO Cd 



^O O 
t— O 



c\j ^a- 
i i 



o c— 
o c— 



CM 

i 



o 
o 



>,0 



CM 

I 



o 
o 



CM 

3 



cr, 
o 






CM 

:1 



O 



C— 



CM 
! 



cr, 
o 



■JD 



CM 
1 



CT> 
O 



E— 
CM 



O 



CM 

I 



O 

o 






CM 
I 



o 



CM 
I 



O 
O 



CM 

I 



CT% 

o 



CM 



Cm 
O 

CO 


h 

-H 

o 

■H 

■a 

2 

to 
o 
■p 

cd 
S 

o 
o 
o 

Cm 

CQ 

C 
O 
•H 

-P 

cd 

cr 
Pd 

•a 



73 

C 

CD 



o 
o 

CD 
OS 



c\J 
ro 

CD 

rH 
-Q 

cd 
H 



C 

o 

•i-4 

cd 
3 
cr 

CD 



in 

CM 



+ 

CQ 



O 

3S 



•P 



CO 

>» 
w 



XI 

CM 






I 

CM 
CQ 



^r 
iO 



o 

a 



CM 

2 



cd 
CM 





«« 


i — i 


i — i 






■\ 


CM 


CM 






cq 


O 


CD 






^^ 


e> 


G 


c 

D 




i_ — 


« 


1 


\ 




CM 






/-% 




• 


•—- V 


-• — v 


cd 


CO 


ro 


cd 


-O 


sr 


>> 


l ! 


CM 


CM 




CO 


"N, 






cr 


3 


CM 


cr 


cr 


CD 


\ 


* — 


(1) 


CD 


^-* 


CM 


• 


^_^ 


^_^ 


H 


IX 


o 


CM 


CM 


-P 


is 


as 


as 








i i 


i — i 


II 


it 


II 


ii 


ii 


C 




CM 


CD 






*v 


s 


D 


P 


H 


H 



in 
^r 

o 

I 

<< 

CM 

■-O 



CM 





«? 








\ 








eq 








^— ' 




ar 




m 




CM 


wP 


en 




• 


CJ 


<r— 




CM 


X 


• 




1 


' — V 


O 


CM 


> V 


•Q 


X 


/-^ 


< 


^=r 


1 1 


w^ 


V, 




w u 


D 


to 


cr 


D 


■s. 


v^ 


CD 




o 


o> 


■» — "■ 


! 


p 


CT> 


P 




■^-■ 


• 




UG 




O 


11 


-P 


1! 






ii 




]] 


tP 




X 




b 




cd 


ca 


\ 


C5 


a 


CO 


H 


p 


a 


cc; 



Cxi 

co 

OS 



Qui 



CM 



cd 



J3 



cd 
ITl 



-Q 

in 



oo 






32 



CM 

o 

+1 



Cm 



Cm 



Ch 



X3 

o 



-H 






c\j 



o 
o 



o 

LA 

■ 

o 
+ 



0) 

o 



yo 






CM 

s 






CM 

I 



cr, 
o 



lo 



• 
CM 



CTv 

o 



m 

! 






CM 

1 



cr. 
o 





O 




o 


in 


• 


« 


o 


en 


1 — 


+ 


+ 


o 


o 


• 


* 


CM 
I 


! 



CD 

in 






o 
o 



0) 



CTi 



CM 



CTi 

O 



VO 

t— 



CM 
I 



CTi 
O 



o 

I 



E— 



CM 
I 



C^ 

o 



LO 
I 



M3 

CM 

I 

o 



o 
I 



r- 

CM 

5 
CT. 

o 



cr 

co 

S- 

o 

cd 

cr 



T3 

co 
x> 
C 
CO 

-p 

X 
CD 

I 

i 
CM 



jo 
cti 



CO 


cd 


^^ 


^r 


^ 


cr 




a) 


+ 


N. • 




H 


a 


i — i 


-p 


eg 






cr 
a> 



CM 



cd 



CM 

cd 

CM 
U3 



D° 



on 



CM 






CM 
X3 

=r 

CM 



on 






D 
CM 



CM 

on 



a3 
CM 

CT 
03 

CM 
2 



CM 



X! 

CM 



m 

I 



on 



as 



cr 
a> 



m 

S3 



CO 
>- 



cd 
on 



cr 

CD 



CM 



cr 



CO 



CO 



n 
en 



cm 



CO 
CM 

O* 
CM 

X 



cd 

CM 



cr 

CO 



S 






cd 



CM 



X 
CM 



a* 
0) 



CM 

2 



CM 



a" 



X 









o 
s 
HP 

o 

-p 

d) 
cd 

CD 
X! 



pa 




3 

bQ 
•H 

Cm 

C 
•H 

-o 

CO 
4-5 

O 
•H 

a 
a) 

-a 

ca 

4J 

c 

cd 

a 

CD 

5- 

3 

ra 

cd 
CD 

««j 

CQ 

•a 
c 

cd 



(35 

-p 

a 

o 

-o 

CO 
CO 

rd 
cq 
cd 



S-, 

CO 
X! 
■P 
S- 
3 
Cm 

T3 

CO 

■a 
e 

3 
o 
Sh 

CO 
X» 

•P 

o 
a 



3 
O 

si 

CO 

CO 
■P 

B 



<m 
Cm 
0) 
O 

o 



cd 
O 

-H 
Sh 
CD 
S 
3 

S3 



CO 



O 

cd 
H 

a 



cd 
S 
•H 

O 

0) 

-a 



o 



CQ 

v| 

O 

o 



Eh 
O 

Cm 

8*. 
O 



(A 

o 



CO 

u 
cd 

co 
-p 

•H 

s 



o 

s- 

Eh 

CiJ 



X> 

t— 

CM 

v| 
< 

pq 
v| 



M 

O 

d-, 

CO 
00 
CO 



M 

o 



O 
CM 

+1 

CO 

U 
cd 

CO 

4^ 

■H 

S 

■H 

H 

b 

O 

J- 

M 



-a 
CD 
-P 

U 
o 
a 
co 
u 

8 

o 
X! 
-P 

cd 
X! 
-p 

Sh 
0) 



CO 



•H 

a 

M 

-P 
•H 

s 



U3 



u 

o 
u 
u 

<0 V 



CM 



o 
CM 



A 



cr> 



N. O 

-P 

a* 
o 



O 



O 
CM 



LO 
+1 

-a 
a 
cd 



t> o 

•v. 

O CM 

+ l 

a 


CO 

4J 

0) 

x> 

CO 

Ch 

cd 
w 

HP 

•H 

s 

■r-t 

H 
O 

Jh 

Sh 



Sh 

O 
(w 

"O 

CO 

4J 

Sh 

o 

a, 

0) 

Sh 

00 

-p 

•H 

s 

•H 



Sh 

o 

Sh 
Sh 



m 

0) 
•H 

C 

>H 

«a) 
"v 

m 

CO 

CO 

Sh 

CO 

> 

co 

Cm 

SA 

Cm 
O 

V> 

o 

H 
Q, 



CO < Q) 

x: v. a) 

-p m w 



a> 

Q. 
CO 

.a 

CO 

a 
o, 

Cm 

o 

c 
o 

•H 
hP 

O 

a 

3 

ch 



CO 
•H 



o 
c% 

o 



M a 



G 

O 
•H 

CO 
■H 

O 

<x> 

Sh 



33 

Except for eqs 11b, 13b, and 14b which had biases of +0.976$, 
+0.820$, and +1.111$, respectively, the bias for every equation given in 
Tables 3-2, 3-3, and 3.4 was less than + 0.6$. 

Precision Summary. For the equations in Table 3.2, the estimated 
relative standard (RSD) limits obtained via propagation of error theory 

for N sys' M 2' a G' N max' and RSE were a11 less than or equal to + 4.5$. 
RSD limits for t« and H< were + 0.2$. 

The precision of the EMG equations in Table 3.2, the width-based 
Gaussian equations, and the calibration curve method of Barber and Carr 
(3D is compared for N sya , M 2 , and a Q in Table 3.5. The results shown 
for the Gaussian equations are valid for 50$, 30$, and 10$ width 
measurements because RSD(W 0>5 ) = RSD(W Q>3 ) = RSD(W Q 1 ). The slightly 
greater imprecision observed for the EMG equations is due to uncertainty 
in the B/A measurement not required for the Gaussian equations. The 
somewhat larger $RSDs for the method of Barber and Carr are probably due 
to interpolation uncertainties (from the calibration curves) unique to 
this method. 

The precision of the remaining CFOMs in Table 3.2 was found to be 
highly dependent on the peak shape. Rather than reporting RSD limits, 
the RSDs for several CFOMs or groups of CFOMs have been plotted vs B/A 
in Figure 3- 1 • 

Other CFOM Equations 

Listed in Tables 3.3 and 3.4 are smaller sets of CFOM equations 
developed only for use in determining if a real chromatographic peak is 
well-modeled by an EMG peak and should not be used to routinely 
calculate any CFOM (including N ) since they are usually less 
accurate, less precise, and more complex than the analogous equations in 



34 



J=) 










« 


O 


c 


T- 


in 


a -p 


• 


• 


m 


• 


GO -H 


^r 


-=f 


st 


OJ 


a S 










i 


+1 


+1 


+\ 


+1 



co 

-p 

•H 

a 

H -r-l 



IS 

G 



G 
O 

g 
u 





O 



OJ 



■o 



o 

+1 



c 

LP. 

o 
I 



o 

CM 

+1 



o 

1 — 

+ 



CM 
I 



o 

o 



to 

+ 



in 



o 
I 








in 




• 




en 




+ 






Cm 


•» 


in 


m 


% 


• 


o 


,-__ 




i 


+1 









CO 










CO 










+ 










CO 










o 










^~v 




cd 






«s 




bO 






*s 




c 






03 




•H 






^-^ 




rH 






c^ 











o 




T3 






s- 




O 






-=T 




£ 






I 

CM 




JU 






fT) 




cd 




i — i 


• 




cu 




on 


O 




CL, 




o 






c 






\. 




•H 




1 


03 
•> — *■ 




>» 




ro 


on 




H 


C 


* 


CM 




B 


O 


o 


+ 




o 


■H 


#■•*», 


m 






-P 


< 


on 




cu 


cti 


N, 






TO 


3 


33 


o 




3 


cr 


x^ 


•— N 






CD 


UJ 


<J 




G 




\ 


"N 




O 




eg 


PQ 




Cm 




*~^ 


in 




CO 




• 


oo 




C 




o 






O 




as 


PO 




•H 




\ 


! 


CO 


-P 




•x 


1 ! 


>1 


cd 




-p 


V 


CO 


3 




« — ' 


CM 


ss 


cr 




00 


Of") 


•\ 


W 




SO 


• 


OJ 






• 


O 


05 


• 




VO 


3b 


-p 


m 










on 




11 


ii 


II 


CD 










rH 




CO 


ctf 


£3 


a 




>» 










03 

3g 


CM 


CM 

SI 



33 

in 









PO 








• 








o 








*-\ 








«*i 








Ss 








pq 








Nw* 








CO 








in 


( — 1 






• 


00 






CM 


=t 








* 






+ 


o 






CM 


+ 






on 

• 


on 






o 


• 






-"V 


o 


CM 


CM 


< 


y-*\ 


V, 


■*N 


■\ 


«J) 


T— 


i — 


03 


v. 


1 1 


1 1 




en 


CM 


CM 


to 


■■ — - 


vP 


O 


• 


X) 


O 


O 


o 


• 






1 — 1 


CM 

1 ! 


1 


1 


rP 


N. 


cd 


X3 




en 


~ 


* 


i 


m 


C\J 


CM 




o 


23 


£ 


« 


J2 


i — i 


i — i 


-p 



o 

IS 

-p 

o 

-p 



■s 

S 



rH 

•H 
B 
•H 

co 



G 

CD 
G 
C 

I 

cd 




B 
O 

-p 

-p 
g 

<D 



b 



H 



o 

-p 



CD 








G 




S 


X) 








•r-l 




TO 


T3 




« 




to 







rH 




to 




t— 




O 


3 




r— 










o 




• 




CM 




rH 


JG 




CM 








H 


CO 




v] 




II 




•H 

5 


CO 








t— 






■P 




1 ^— 




• 




TO 


G 


■ 


• 




o 




-P 





u 


o 




•— V 




■H 


S 





' — ^ 




«fl 




a 





.G 


■etl 




~N. 




•H 


G 


-P 


\ 




03 




rH 


3 


G 


m 




■* * 






CO 


3 










G 


CO 


Cm 






-P 




o 







v l 




cd 




G 


a 


T3 










G 







o^ 




a* 







rr- 


•a 


o 




O 






« 


a 


• 


*> 





■ 





O 


3 


« — 


vO 


CM 


»a 


x: 


s~*. 


O 




c— 




c— 


-p 


eti 


G 


o 


• 


+1 






"x 




-p 


CM 




CM 


M 


03 









O 




O 


^^ 


JD 


CO 


V| 


-P 


v| 


CM 




-P 


G 


1 


o> 


X— 


A 


• 


O 


O 


• 


o 


• 




CO 


G 


Q, 


o 


• 


o 


O 


-P 




TO 


*^-v 


v— 


-<"> 


O 


G 


-a 


cu 


< 




< 


\ 


cu 


i— i 


G 


\ 


II 


^ 


o 


s 


3 


G 


03 




03 


+J 





o 


O 


-^S 


K — 






G 


JG 


O 




. 




G 


3 


TO 




v| 


o • 


v| 


O 


TO 




-C 




^— V ^- 




Ct, 


cd 


TO 


o 


o> 


=c • 


a. 







-P 


■rH 


1 — 


s on 


K 


(J 


a 


G 


JB 


• 


03 


• 


o 







3 


« — 


w 


1 


CM 


en 


-H 






G 






• 


O 


•* 


G 


-P 3 


G 


II 


o 


•H 


CO 


O 


Cd hO 


O 




■**-» 


Cm 


o 


Cm 


•rH 


Cm 


o 


«a 


Cm 


• 




-tf*V Cl, 




io 


\ 





CM 


«* 


CT> 


S* 


x. 


Q3 


o 




o 


• C 


O 


o 


^—^ 


o 


V 


CM 


m -h 


Z± 


-p 


TD 


H 


en 


+ 


, c 


+ 


G 


C 


cd 


• 




+1 3 




O 


Ctf 


o 


o 


* 


o 


•« 


Cm 




•H 


* — * 


•sS. 


a jc 


a«. 






G 


< 


in 


O TO 


in 


■a 


•» 





■x 


• 


G 


■ 


<v 


on 


a 


oq 


o 


<m 


^ — 


•p 


• 


3 


t— ^ 


1 


> 


1 


G 


o 


s 






TO G 




O 


3 




v| 





3 





Q. 








G 


W3 O 


G 





K 


CO 


o 


cd 


S o 


cd 


G 


CC 





■ 


TO 


G D 


00 


TO 


-P 


o 


« — 


-P 


N. 


-p 


4J 




cd 




•H 


H 


■rH 


•H 


G 


^H 


G 


s 


G 


s 


a 


O 


Oh 


O 


■H 


° L ~ 


■rH 


•H 






Cm 


rH 


■rH H 


rH 


rH T> 




rH 






TO 







TD 


CO 


T3 


G 


■H 


G 


G -P 


CD 


a 


•H 


O 


o j.: 


O 


O G 


W 


•H 


rH 


G 


-p 


G 


G O 


cd 


o 


cd 


G 


G 


G 


G Q, 


m 





> 


a 


cx, o 


eg 


K) 


cd 


■o 


a 


o 


T3 .p 





Cm G 



35 





























G 






J3 
























cd 







W 

Q -P 


sr 

« 


t— 

* 


• 


CM 






CM 










-1 
•H 


00 

o 




os s 


o> 


t— 


o> 


OO 


T3 


-a 


O 


o 

is 








a 


si 




rH 


+1 


+1 


+1 


+1 






+1 








CO 


G 




























G 


cd 




















o 








CD 


x; 




















J- 1 








C 
C 


-p 




















CD 








cd 


G 




























a 


CD 




£> 






















cd 


H 
rH 




CQ 


























cd 




-P 

•H 

a 

H -H 

CD rH 




o 






o 

LO 


LO 




CD 
-Q 








C 


a 

CO 






CM 

+ 






* 

LO 

+ 


CM 

+ 




■a 

H 
3 




• 










G 

we. o 

U 

g 

0) 


o 

CM 
+1 


o 
1 

1 — 1 

o 

+ 

LO 


o 

CM 
+1 


O 
CM 

+i 


O 
1 


O 
1 


CD 
LO 

• 

O 
+1 


o 
X! 

CO 
CO 

-p 

c 

CD 

s 

CD 
G 
3 
CO 

cd 

CD 

B 


• 

CD 

-G 
■P 

3 
Qh 

■a 

CD 


CM 

v| 

o 

< 

X 

a 
v| 




CM 
II 

O 

cq 

-p 
cd 

a*. 


rH 

^H 

* 

CO 
JJ 
•H 

a 

•H 
rH 

G 
O 
G 
G 
CD 


















LP 


T3 


o 




CM 




















a 


G 
3 


^— 


• 
>j3 


cn 


CD 

si 








o 












<c 


O 

G 


o 


C— 


M 


4J 




















"V. 




J--- 1 


CM 




M 


















cq 


CD 






o 


O 


cd 






P3 












■ — 


n 


CQ 


v| 


-p 


CM 


bQ 






















T3 








a 






PO 














+5 


C 


* — 


c^ 


/\ 


■H 














i — I 


• 


o 


O 


• 


o 




H 

0) 

■§ 






CM 
t- 
1 
CM 










OO 


CO 

-p 

G 
CD 


c 

rH 


a 

CD 


o 

■a) 


II 


C3 


2 






in 










l 


CD 


3 

O 


G 
O 


cq 





+^ 


3 




1 ! 


o 










LO 


G 

3 


£3 
co 


CJ 


v| 


o • 


G 
O 


CD 




t— 


>a! 










• 


CO 




XS 




^^ T - 


En 


a, 






s^ 










o 


CO 


CO 


O 


c^ 


«tj . 








o 


CQ 










*■* 


CD 


-P 


•H 


T — 


"N. 0O 


« 


G 
















■t! 


a 


G 


JS 


• 


PQ 


O 


•H 




1 


X) 










X 

CQ 


CO 


■H 


3: 


T_ 


w (D 
G 


CM 


>> 




in 


, , 










— 


• 


O 


•» 


G 


+> 3 


!! 


H 


c 




^ 










LO 


o 


•H 


O 


O 


cd tsO 




C 


o 


o 


♦ 












**"'* 


Cm 


C— 


Ch 


•H 


v^O 


o 


•H 




co 










+ 


<aj 


Cm 


• 




»* bi 


b 




.p 


«s! 


LO 












X 


CD 


*— 


'as. 


LO 


X 


0) 


Cfl 


\ 












CM 


03 


O 




o 


• c 


o 


CO 


3 


cq 


o 










LO 


* — 


O 


V 


• 


=r -h 


+J 


3 


a* 














1 








CM 


CM 






CD 


i — i 


< 




| 






o 


-a 


H 


lO 


+ 


, s 


G 


G 
O 




X 

CM 


X 

CQ 




LO 






«) 


c 
cd 


cd 
o 


o 


H 


+l § 


O 

Cm 


Cm 

CO 
C 
O 
•H 
•P 




LO 

O 

IS 

cc 


CM 

00 



U-l 


CO 

>> 


o 

«si 

* — ' 
LO 


CM 
X 

* — i 
CM 

a 
b 


CM 

X 

i — i 
CM 

a 


X 
PQ 

1 * 

-o 


LO 

o 

3 


■H 
S-, 
CD 

s 

3 

3 


CQ 


o 

T 

CD 


3 S3 
O CO 
G 

C(H CD 

> 
CO G 
CD a 


CD 
-P 
G 
O 


cfl 




■P 


X 


CO 




D 


1 








G 


bO O 


a> 


3 




PO 


CM 


a 
s 


OJ 





1 


—J 

a 


~ 


CO 


o 


cd 


G 

cd o 


G 


id 




00 


O 


CM 

05 


LO 


cd 


-O 


D 


K 
-P 


CD 
O 

cd 

rH 

O. 


•^ 


CO 

-p 


G D 


CO 
-P 








3 


-P 


O 

3= 


CM 

X 


CM 
2 


cc 


a 
o 


G 
O 


•H 

S 
•H 


G H 

O » 


•H 

a 

•H • 


oo 




II 


1! 


li 


i — i 


i — i 


-p 






Cm 


^H 


•H M 


rH T3 






















H 






CO 


CD 


CD 
H 




CO 


CO 


£3 


II 


IS 


n 


II 


•a 

CD 


cd 
S 


-a 

■H 


G 
O 


■H CD 
O A 


G -P 
O G 


-Q 




>» 














w 


•H 


H 


G 


CD -p 


G O 


cd 




CQ 


CM 


CM 


a 


CO 


£3 


'J 
-p 


cd 
cd 


CD 

CD 


cd 
> 
J3 


G 

o 


G 

C1h O 
-O +> 


G Q, 

aq cd 

CD G 






36 



Table 3.5. Precision 3 - Comparison of Three Graphical Methods for 
Estimating N , M 2 , and ° G 



method 
Gaussian eqs 



sys 



+ 2.0 



M- 



+ 2.0 



+ 1.0 



empirical eqs, 
Table III, 
this work 



+ 2.5 



+ 2.4 



± 2.0 



calibration curve 
method, reported 
previously (3D 



+ 5 



+ 3 



± 3 



a,. 



Reported as percent relative standard deviation (%RSD). 
Precision of equations estimated via error propagation, usins 
data of Table 3.6. 






rH 


r4 


H 


a 


H 


CO 


u 


cti 


CD 


CD 




O, 


> 






CD 


<4-l 




CO 


O 

c 


Cm 
O 




o 


-P 


u 


•H 


a 


o 


CO 


a; 


Cm 


•H 


TD 




O 


c 




a) 


CD 




S-, 


a. 


>> 


Q. 


CD 


L, 




T3 


■P 


CD 


c 


CD 


n 


•H 


i 


£-i 




i 




>> 


>» 




rH 


GO 




H 


ffl 


• 


rc! 




* — . 


•H 




ra 


-P 




2: 


c • 


ij 





CD C— 


C3 


[& 


W • 


CD 





co m 


Q, 


~— ' 


CD 1 

C\J 




-P 


03 • 




«rl 


•h on 


C 


<-, 




o 


CD 


*-> CO 




5 


• CD 




r M 


■P n 


c 





CD cvj 


o 




H 


■H 






w 


co 


- c 


■r-l 





CM-H 


O 


!m 


•y^ 


<y 


3 


■a 


Sh 


0Q 


CO 


a. 


■H 


» -a 




Cm 


CO 3 
COO 


M-t 


O 


z c 


o 


•H 

Si 


~-^ -H 




Q, 


CO CO 




cd 


S -H 


Q> 


L< 


O 


O 


b0 Cx, T3 


a 


O 


c 


CD 


-P 


cd 


X) 


ci! 




a 


a 


m CD 


CD 





a> a. 


a. 


u 


X! cd 


cd 


a 


-P si 


a 





O CO 



<D 
Eh 

to 

•H 



_. . . 






38 



^ 



sj- ro - Q_ 
^ IS I- 01 




CD 

cvi 



in 



C\J 

c\i 



o 



CD 



ro 



< 

CD 



O 

ro 



in 

CM 



_J__ 

O 
C\J 



in 



as a % 



in 









39 



Table 3.2. However, the accuracy and precision for MU, a Q , t Q and t at 
a = 0.3 and 0.5 are still sufficient to permit peak modeling decisions 
to be made. 

Discussion 
Detailed Discussion of Precision 

Shown in Table 3.6 are the precision data for t p , W , and (B/A) 

It EL 3, 

used in this study. The RSD results were obtained by converting 
previously reported raw precision data (32) to the form appropriate for 
error propagation analysis for the conditions specified in Table 3.6. 
Data excluded in the previous study for (B/A - 1) were also excluded in 

CI 

this analysis. The RSDs of t R , W & , and (B/A) for individual peak 
shapes (at a given peak height fraction) were averaged as done 
previously, thereby implicitly assuming the independence of the RSDs on 
peak shape. 

The RSDs for t R , W Qf and (B/A - 1 ) for individual peak shapes were 
originally reported relative to a Q , a Q , and (B/A - 1) , respectively. 
Multiplication by o Q /t R , ° Q /^ a , and (B/A - 1) /(B/A) converted them to 
the appropriate form. 

Since the RSDs of t R , W & , and (B/A) were assumed to be independent 
of peak shape, intuitively it might seem that this should also be true 
for the RSD of any calculated CFOM. This is not the case, however. For 
one group of CFOMs (N sys , M £l o Q| t Q , N max> RSE, and M,), a slight to 
moderate variation in their RSDs with (B/A),, was observed. This can be 
explained by examining the random error propagation in the general 
empirical N equation, 

N sys = C 1 (t R /W a ) 2 /[(B/A) a + C 2 ] (3.9) 



■ 



40 



Table 3.6. Suggested Chromatographic Measurement Conditions and 

the Resulting Precision (%RSD) Achieved for t„, W , and (B/A) 

n a a 

Conditions 

1. Chroma togram recording rate: 1 cm/o G (W Q < _> 4.3 cm) 

2. Ruler resolution: +0.2 ■ 

3. Minimum retention distance, (t R ) . : 10 cm 

4. Minimum peak height, (hp) min : 10 cm 

Results a ' b 

CFOM (a = 0.1) (a = 0.3) (a = 0.5) 

fe R +0.2 identical identical 

w a ± 1 -° ± 1.0 + 1.0 

(B/A) +2.0 +2.5 +3.0 



Data obtained from reference 32 and subsequently converted (see 
Detailed Discussion of Precision) for a = 0.1, 0.5 — results 
interpolated for a = 0.3. 

%RSD(t R ) rounded to nearest 0.1%; %RSDs for W , (B/A) rounded 
to nearest 0.25%. 



41 



Assuming negligible covariances, RSD(N ) is given by 

sys 

RSD(N sys ) = {4[RSD(t R )] 2 + 4[RSD(W a )] 2 + 

1 2 

RSD[(B/A) a ] 2 [(B/A) a /((B/A) a + C 2 )] 2 } 1/2 (3.10) 

3 4 

Even when terms 1-3 in eq 3.10 are constant, RSD(N „) will vary 

sys 

somewhat with (B/A) a because of term 4. Clearly this variation will be 

greatest for -(B/A) a < C 2 < 0. Additionally, as C 2 ■+ -(B/A) , 

RSD(N gys ) -*• ». For C 2 =0, RSD(N sys ) is essentially independent of 

(B/A) a . Finally, for C 2 > 0, a negligible to slight variation of 

RSD(N s ) with (B/A) a may be observed, depending on the magnitude of 

terms 1 and 2 relative to the product of terms 3 and 4. Except for eqs 

1 and 2(a,b) in Table 3-4, the RSD limits for N , M OJ a r , t P , N 

sys ' d ' h ' b ' max ' 

RSE, and M., calculated via equations in Tables 3.2, 3.3, and 3.4 varied 
by less than 0.5% for 1.00 <. B/A <. 2.76. 

The remaining CFOMs (t, t/o q , RPL, M,, M 4 ,Y s ,Y e ) comprise a second 
group whose RSDs are moderate to strong functions of peak shape as 
Figure 3-1 shows. In every instance the imprecision is largest for the 
least asymmetric peaks and smallest for the most highly skewed peaks. 
Analysis of the error propagation equations show that one or more terms 
within the equations get very large as the peak shapes become symmetric. 

Why Measure at 10% Peak Height? 

For reasons listed below, the recommended CFOM equations in 
Table 3.2 are based on (in addition to t R ) the measurement of W and B/A 
at 10% peak height rather than at other peak height fractions such as 
50%, 30%, or 5%: 



•--*--—-----■ 



42 

1. Examination of Tables 3.2, 3-3, and 3.4 shows that many CFOM 
equations at 10$ are olearly superior to the corresponding ones at 30$ 
and 50$ in terms of 

a) precision (lowest RSD limits) 

b) widest working range for equivalent accuracy (e.g., N ) 

c) simplicity for M~ (path a) 

2. The N s equation at 10$ peak height is more accurate for Gaussian 
and near-Gaussian peaks than other N equations developed at 50$, 30$, 
or 5$. For example, at B/A = 1 the relative error was +0.6$, +10.0$, 
+2.0$, and +2.5$, respectively. 

3. In a previously reported graphical measurement study (31,32), 
statistically significant positive and negative biases were detected in 
the measurement of A Q 5 and B Q 5 , respectively, resulting in a 
consistent underestimation of (B/a) q 5< No such biases were detected 
for Aq ^ and B Q «, and only a slight underestimation was observed for 
(B/A) Q> J. 

4. It is likely that RSD(W 0#05 ) > RSD(W 0<1 ), since in going from W Q 1 to 
W 0.05 the ma S ni tude of the slope of the peak (on either side) decreases 
much more rapidly than the peak width increases. Thus the precision for 
the 5$ CFOM equations would be poorer [assuming RSD(Wq Q5 ) contributes 
substantially to the total uncertainty]. 

5. Superior resolution between overlapping peaks is required for 
measurements at 5$ peak height than at 10$. 

6. As exemplified in Figure 3.2 for N , Gaussian CFOM equations based 
on width measurements at 10$ are much less inaccurate (though still 
exceedingly in error) for asymmetric peaks than the corresponding 
Gaussian equations at 50$ (shown) and 30$ (not shown). That is, the 
slope of the RE vs. (B/A) a (shown for a = 0.1) plots is smaller; thus 
the approximate RE correction function in the denominator of eqs 3.2 and 
3.5 (text) will be less sensitive to the measurement imprecision of 
(B/A) a ). 

7. The sensitivity of the relative error (RE) correction functions to 
the (B/A) a measurement imprecision is only slightly lower at 5$ than at 
10$ peak height (see Figure 3.2) and is insufficient to warrant CFOM 
estimation at 5$. 

6. As seen in Table 3.7, the RSE can be calculated much more accurately 
using width measurements at 10$ than at 50$, 30$, or 5$. In addition, 
the precision is much better (lower RSD limits) at 10$ than at 50$ or 
30$, and is comparable to that at 5$. 

9. The empirical asymmetry factor measurement, B/A, was introduced at 
10$ peak height rather than at 50$ or 30$ because peak tailing is much 
more apparent at 10$. Since then almost all empirical measurements of 
asymmetry have been reported at this peak height fraction; these data 
will be of little value in later years if the B/A peak height fraction 
is redefined. 






c 


V 




o 


>% 




m 


X3 


as 




<D 






<n 






cd 


-o 




ja 


CD 
El 




m 


JD 




c 


3 




o 


a 




•H 


£ 


• 


-P 


H 


CQ 


cd 




J* 


3 




cS 


CT 


<D 


0) 


CD 


G 
P 


a 


G 




o 


rt 




0) 


•H 


X) 


3 


H 


C 


(D 


CQ 


rt 


Js: 


3 




tn 


CT3 


' V 




a 


w 


-a 




p 


a 


0) 


c 


cd 


CD 


CD 




u 


g 


H 


-G 


5 


cd 


-p 


G 


a) 




3 


X5 


Cm 


CO 


•i-l 


o 


CO 






<D 


Li 


-p 


g 


O 


c 




Cm 


3 


X! 




o 


i> 


**■■* 


o 


•H 


* 


cd 


3 


cr 


+-•> 




CD 


cd 






rH 


M 


*• 


O. 


tt) 


C\J 




a; 


• 


c 


a. c^ 


■H 




CD 


G 


X) 


H 


o 


CD 


-Q 


Sh 


P 


m 


G 


cO 


IH 


<u 


•H 


s -" ' 


cd 


■a 


C 


> 


C 


O 


■H 


■m 


•H 


-P 




43 


CO 




CO 


H 


CD 


3 


CD 


Si 


cr 


03 


4-1 


CD 



OJ 



en 

CD 

aO 
■H 
En 



44 



d 



5 



in 

o 
o" 



>- 

00 




1 



O 
O 
CO 



i i ■ » 



O 



J 1 1 1 1 I I u 



O 

o 



o 

lO 



aoa*d3 !N30d3d 



in 



- C\J 



< 

m 



io 



45 



Table 3.7. Comparison of the Accuracy and Precision of the 
fiSE Equations at a = 0.05, 0.10, 0.30, and 0.50 



equation 



RSE = 1.04 [(B/A) 0#05 r 2 - 00 



RSE = 0.99 C(B/A) Q#10 ] 



-2.24 



RSE = 0.926[(B/A) 0>30 ]- 3 ' 11 
RSE = 0.913[(B/A) 0>5Q ]- 4 - 33 



#RE limits 


% RSD 


-5.0, +4.5 


+ 4.0 


+ 2.1 


+ 4.5 


-7.5, +2.3 


+ 8.0 


-9.0, +4.5 


+ 13.0 



" 









46 

10. It is easier to mentally compute \Q% of an arbitrary peak height 
than 50%, 30%, or 5%. 

Taken collectively, the above arguments indicate that the best CFOM 

estimation is obtained from graphical chromatographic measurements at 

10% peak height. 

General Aspects 

Preliminary mode ling of experimental peaks . Chromatographic peaks 
should be examined for their resemblance to Gaussian, EMG, or other peak 
shapes, first by visual inspection and then from the asymmetry factor 
measurement. In the unlikely event that B/A = 1, the validity of the 
Gaussian model can be checked by comparing the measured peak width 
ratios W Q>5 :W q.3 :W 0.1 to the theoretically predicted ratios 0.5487 : 
0.7231 : 1. For B/A 2 1.09, the validity of the EMG model can be judged 
by the agreement of values of o Q , M 2 and/or x, and t Q determined from 
both B/A and W a measurements at a = 0.1, 0.3, and 0.5 (see Tables 3.2, 
3.3, and 3.4). 

For slightly asymmetric chromatographic peaks, the assignment of 
peak shape models may be ambiguous due to the imprecise measurement of 
B/A (e.g., Is a peak with B/A = 1.03 ± 0.02 Gaussian?). Insofar as 
accuracy and precision are concerned, does it matter if EMG-based 
equations are used with Gaussian peaks or vice-versa? As seen from 
Table 3.2, the EMG based equations for N , M 2 , and a Q are accurate to 
within ±1.5%, ±1.5%, and ± 4%, respectively, over the asymmetry range 
1.00 < B/A < 1.09. Figure 3.3 shows the accuracy of the Gaussian based 
equations (a = 0.1) over this same range. Clearly, little error in the 
estimation of M 2 , N sys , and a Q will result from peak model 
misassignments at low asymmetries (1.00 < B/A < 1.09) due to B/A 






T3 






a 




• 


<a 


o 




3 




■* 






jj 






a 


~ 




3 


as 


O 


.p 




o 






CD 


c 




40 


o 




cd 






.H 






Q. X 






CD 




■» 


CO 




a 


03 


• 


o 


J3 


CO 


•H 




M 


40 


CO 


03 


03 


c 


CD 


■H 


o 


Q. 


> 


■H 




© 


.p 


X) 


T3 


CO 


CD 




3 


S 


-o 


CT 


CD 


S-. 


CU 


a 


cd 




CO 


-o 


a 




C 


cd 


>> 


cd 


•H 


^H 


-P 


CO 


■P 


CO 


n 


43 




3 


bO 


C 


03 


•H 


03 


o 


H 


•r-l 




CO 


CO 


G 




CO 


o 


X) 


3 


u 


c 


03 




cd 


a 


Jj 






c 


rH 


CD 


0) 


cd 


si 


a 


CD 


-p 


o 


-o 




s 


•H 


c 






•H 




G 




rH 


O 


G 


cd 


<w 


O 


G 




!h 


-p 


CO 


G 


B 


-P 


d> 





G 







CD 


CD 




a 


> 




CD 


■H 


'O 


G 


■P 


C 


3 


03 


O 


CO 


H 


o 


cd 


CD 


CD 


CD 


SB 


CO 


S 


ro 






i^O 






a) 






G 






3 






bO 










48 




J i_ 



' ' i l i I I I i — i — L 



Q 

o 



10 



o 

in 



if) 
cvi 



o 
d 



U0UU3 !N30d3d 



m 

CM* 

i 



O 



CD 
O 



ro 
O 



< 
m 



o 
o 












49 

imprecision, though for B/A > 1.04 the EMG equations are more accurate. 

Furthermore, as seen from Table 3.5, the change in the precision of CFOM 

estimation resulting from peak model misassignment would be less 

than 1$. 

Double-checking the results . The universality of the relative 

error approach (introduced in the Derivations section above) was checked 

by the independent variation of t Q and Oq over the t Q / a G range of 10 to 

5000. The generality was confirmed by identical statistical results 

(£RELs, etc.) over a given B/A range (e.g., 1.09-2.76) for all CFOMs 

calculated from this approach. 

As an additional check on the experimental work, M 5 and N_,,_ were 

£~ sys 

calculated by statistical moment analysis (23), using the same to search 
algorithm as before. The agreement among the true, moment, and manual 
values for both M 2 and N was within + 1$ for 1.00 < B/A < 2.05 with 
very little bias present in either approximation. At higher asymmetries 
(2.05 < B/A < 2.76), the manual values remained within + 1.5$ of the 
true values, but the corresponding moment values showed a significant 
positive bias ranging from +1.5$ to +5.5$. An even greater bias of 
+4.4$ at B/A = 2.05 ( T / a G = 2) reported elsewhere (23) was attributed to 
arbitrary data truncation. This was probably the source of bias in our 
statistical moment method as well, but regardless of the source of bias 
the same conclusion may be drawn: at high asymmetries (2.05 < B/A < 
2.76) the manual CFOM equation method is more accurate than the moment 
method. 

Working range of the equations . The EMG-based equations in 
Table 3.2 were expected to be accurate over the asymmetry range used for 
the least squares curve-fitting of the f(B/A) approximations. Thus, 






50 

except for eq 8, the accurate working range was thought to be 1.09 < B/A 

< 2.76. Nevertheless, eqs 1, 2(a,b), 6, 7, and 10 allow accurate 

estimation of N , M 2 , t Q , N fflax , and M 1 , respectively, for Gaussian 

shaped peaks (B/A = 1.00). Although somewhat surprising, this is 

explained by the near convergence of these EMG equations to Gaussian 

ones when the substitution B/A = 1.00 is made in the former. The N 

sys 

equation in Table 3.2, for example, becomes 

N sys = l8.53(t R /w 0<1 ) 2 (3.11) 

which is within +0.6$ of the Gaussian formula 



Vl = 18 ' 42( V W 0.1 )2 - (3.12) 



CFOM equations could have been developed for asymmetries greater 
than B/A = 2.76 (T/a Q = 3), but a sacrifice of simplicity, accuracy, or 
both would have been required. More importantly, however, it was felt 
that nearly all peaks reported in the literature exhibit B/A's < 2.76. 
Indeed, a chromatographic system producing peaks with B/A's > 2.76 is 
operating at a relative system efficiency of less than 10%. 

When the EMG N s equation was tested for asymmetries higher than 

those for which it was developed, the $RE varied between -1.5$ and -10$ 

for 2.77 < B/A < 4.00, compared to the %RE range of +70$, +110$ for the 

Gaussian 10$ equation (N,, ). 

w 0. 1 

Real versus ideal CFOMs: column characterization . Given that peak 
asymmetry is (almost) always present in any real chromatographic system, 
N sys' M 2' and fc R ^present the experimentally observed chromatographic 
efficiency, peak variance, and retention time, respectively. The 






51 

corresponding CFOMs N max , a^ , and t Q represent idealized 
chromatographic parameters which would describe the system if all 
sources of asymmetry could be eliminated. If all or nearly all 
asymmetry is extra-column in origin, then for a given set of conditions 
« fflax i a c » and fc G are valid descriptors of the efficiency, band- 
broadening, and retention characteristics of the column. 

Pluralism of the method . As might be surmised from Table 3.2, 
there is more than one way to calculate several of the CFOMs. The 
variance, for example can be calculated via eq 2a from measurements of 

w 0.1 and B/A or via e< 3 2b frora fc R and N S ys ( e 3 1 )- Generally, CFOM 
estimates via the "b" equations are simpler, faster, equally precise, 
but less accurate than estimates via the "a" equations. This trade-off 
of accuracy for simplicity and speed is slight, however; in most cases 
much time can be saved with little sacrifice in accuracy if the "b" 
equations are employed. 

Only the simplest and most accurate methods are given in Table 3.2. 
Therefore, while N could be calculated from its components t R , a Q , 
and t (see N equation in Table 3-D, this method was not reported 
since it would be much more time-consuming, tedious, and in all 

likelihood less accurate and less precise than the N _ equation in 

sys 

Table 3.2, 

Usefulness of RSE T RPL . The relative system efficiency (RSE) and 

relative plate loss (RPL), two new parameters defined in Table 3.1, are 

dualistic CFOMs. First, they can be interpreted intuitively as 

N sys /N max and (N max " N sys )/N max> respectively, with a corrective 
retention factor (t Q /t R ) 2 applied. Alternatively, RSE and RPL can be 
viewed as the relative contributions of symmetrical ( a Q 2 ) and 






52 

asymmetrical (t ) band-broadening processes to the total system band- 
broadening (M 2 , the total variance). If (t G /t R ) 2 can be neglected 
because of its nearness to unity, the former intuitive expressions for 
RSE and RPL become particularly useful. For example, the best possible 
efficiency, N max , can be related very simply to the true chromatographic 



efficiency, N sys , by 



N max = *sys /RSE (3.13) 



In fact, this approximation is good to within + 2% with an RSD limit of 
+ 5% (for 1.00 < B/A < 2.76) whenever N „ (calculated via eq 3.13) > 
4000. Thus, eq 3.13 can serve as a useful estimation of S__„ for 

HldA 

moderate to high efficiency chromatographic systems. 

Although B/A, RSE, and RPL are mutually interdependent (i.e., once 
B/A has been measured RSE and RPL may be calculated), the specification 
of RSE, RPL, or both in addition to the reporting of B/A greatly 
enhances the qualitative description of a chromatographic system. Thus, 
while B/A = 1.30 indicates that a given peak is asymmetric, the 
corresponding RSE = 55$ or RPL = H5% provides a clearer indication of 
the actual efficiency and how much room for improvement exists. 

Figure 3.4 shows the exponential-like relationship of RSE and RPL 
with B/A. Chromatographic systems with asymmetries of 1.00 and 1.10 are 
operating at much different relative efficiencies, while two systems 
operating at B/A = 2.00 and 2.10 are realizing nearly the same relative 
efficiencies. 

Alternative derivations . Two other approaches for deriving CFOM 
equations were not as successful as that already described. In the 
first attempt, a modification of the Carr graphical method, three 
universal calibration curves were approximated by linear or quadratic 



53 



RPL 




RSE 



B/A 



Figure 3.4. Relative system efficiency, RSE, and relative plate loss, 
RPL, for ideal and skewed peaks. 






54 



polynomials. This approach yielded equations for a Q , x , t Q , and the 
remaining CFOMs in terms of W Q>1> B/A, and t R , but was unsatisfactory 
for three reasons: a) an accurate but simple approximation of t/a„ in 

u 

terms of B/A cannot be obtained for the range 1.00 < B/A < 2.76 because 
the relationship between them changes at B/A = 1.36 from a decidedly 
nonlinear one to an almost perfectly linear one; b) the errors 
introduced by the polynomial approximations tend to accumulate slightly 
rather than cancel; and c) the equations derived for all the CFOMs 
except x, o G , and t Q are extremely unwieldly, e.g., eq 3.14 below, 

N sys = (t fi /w 0>1 ) 2 [ g (B/A)/h(B/A)] (3.14) 

where g and h are second degree polynomials of B/A. A variation of this 
same approach in which B/A was substituted for the original abscissa 
x /°q in the latter two calibration curves made little difference. 

A second attempt, a variation of the relative error {RE) approach 
utilizing (B - A) as the approximation G in eq 3.5 for t, failed 
because the simplest f(B/A) approximation required for sufficient 
accuracy was too complex. 

CFOM units. For ease of interlaboratory comparison, all non- 
unitless CFOMs should be reported in time units; if units of length are 
chosen instead, the recorder chart speed should be specified to permit 
conversion to time units. 






55 

Conclusion 
Superiority of EMG-based Equations . Although Gaussian-based 
equations are somewhat simpler than their EMG counterparts in Table 3.2 
(compare eq 3.11, text with eq 1, Table 3.2), the EMG equations are 
clearly superior because with comparable precision (see Table 3.5) they 
are equally accurate for Gaussian (or near-Gaussian) peaks and 
considerably more accurate for skewed (EMG) chromatographic peaks. 



CHAPTER 4 
CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY 



Introduction 
In this chapter, the focus is on eliminating the confusion 
surrounding the limit of detection (LOD) in chromatography. Some prior 
knowledge of the LOD concept is assumed. This discussion primarily 
applies to both gas and liquid chromatography using concentration or 
mass sensitive detectors whose output is measured as peak height, though 
two sections (Eliminating Mistak en Identitiaa and Using the Correct Units 1 
apply to analyses using peak area as well. Ideal linear elution, i.e., 
Gaussian peak profiles, is assumed though moderate deviations can be 
tolerated,, 

Identifying Current Problems 
Literature Survey Results 

Initially, to determine the sources of discrepancy in 
chromatographic detection limits, we conducted a limited survey of 
analytical textbooks, chromatographic monographs, and the primary 
chromatographic literature. This survey revealed two mistakes of 
omission, as well as four major sources of discrepancies. All six are 
summarized in Table 4.1. The first two problems are blatant omissions 
which discredit the work reported, at least to some degree. However, 
they can be eliminated if more attention is given during manuscript 
preparation, and thus may be dismissed without further discussion. The 

56 






57 



>> 

ft 

CIS 














P 








. 


• 

a 




■ 
CO 




o 

B 


CD 
G 


o 




T3 


<D 




•H 




cd 


CD 


-p 




<D 


> 




CO 






5 


cd 




M 


■H 




>» 




G 




a 




CO 


60 




rH 




cd 


Q 


o 




0) 






cd 




.G 


o 


h 




G 


co 




q 




■P 


J 


jO 




-P 


ca 




cd 








o 




co 


3 






• 


G 


CD 


C 










CD 


C 


CD 


x: 




CO 


a 




o 


O 


a 


-P 


•H 




cd 


o 




cd 


•H 


p 








3 


j 




G 


-P 


cd 


£j 


-P 










P 


cd 


G 


o 


ft 




CO 


CD 






o 






CD 




•H 


J3 




c 


•H 


G 


CD 


O 




CO 


+3 




-H 


Cm 


O 


> 


g 




>> 








•H 


•H 


cd 


O 




H 


G 




T3 


-P 


-P 


jC 


o 




cd 


o 




CD 


co 


cd 








C 


Cm 




CO 


3 


G 


CO 


^^ 




cd 






3 


1-? 


■p 


G 


C5 






C 








c 


CD 


o 




0) 


O 


CO 


CO 


o 


CD 


-P 


J 




o 


■H 


CD 


-P 


c 


CJ 





*~— ' 


C 


cd 


-P 


•H 


a 




a 


a 




O 


G 


•H 


O 


CD 


G 


o 


cd 


C 


•H 


4J 


a 


G 


o 


O 


o 


G 


O 


CO 




•H 


cd 


G 






cd 


■H 


CO 


x; 


Cm 


a 


o 


© 


Cm 


a 


+5 


-.H 


bO 


CD 


CD 


o 


rH 


O 




O 


a 


3 


X3 


G 




-P 




o 


a) 


o 


O 




o 


G 


-P 


CO 


•H 


-p 




G] 





CO 


CD 


■H 


4-> 


jC 


0) 


Cm 


-4-> 


C 


■H 


43 


rH 


•H 


a 


X3 


O 






X5 


-P 




G 


cd 






*• 


•P 




o 


J3 


3 


G 


Ch 


co 


-o 


a 


Cm 




-P 




bO 


o 


eu 


CD 


o 


o 


SS 


■H 


c 


O 




M 


■P 






•p 


3 


•H 


-p 


-p 


cd 


G 


~ 


60 


■H 






cd 


•H 
S 


4-S 


O 


T3 


CD 


3 


■a 


■a 


s 


CO 


ft 


CD 







CD 


0) 


o 


■H 


•H 





-P 


G 


T3 


CO 


-p 


G 


rH 


a 


t, 


Eh 


3 


CD 


3 


G 


X3 











O 


CO 




O 


O 


CD 




-P 


ft 


CO 


3 


CD 


a 




,G 




o 


CD 




Cm 


G 


CD 


rH 


-P 




c 


U 




G 
O 


a> 
3 


G 


cd 

G 


X! 




CD 


CD 




O 




CD 


CD 


+J 




1, 


G 






CO 


G 


> 


•H 




CD 


a) 




CO 


rH 


CD 





2 




3 


3 




cd 
3 


CD 

■a 


3 


CO 


4) 




co 


CO 






o 


CO 


-p 


S 




.p 


■P 




4J 


e 


-P 


ca g 


(1) 




•H 


■H 




a 




•H 


.c o 


t-i 




a 


e 




CD 


a 


a 


-P Cm 


.o 




■H 


•h 




o 


o 


•H 




o 




rH 


H 




G 


j 


rH 


co -a 


g 




g 


C 




O 

o 


>> 


G 


-P CD 
O -P 






o 


O 






G 


o 


cd a 






•H 


•H 




Q 


cd 


-H 


Cm 3 


• 




*3 


-P 




O 


G 


-P 


Cm O 


* 







O 




J 


•P 


O 


CD O 


• 




CO 


CD 






•H 


CD 


Q 


^T 




-p 


-P 




(D 


£) 


■P 


CD cd 






<D 


CD 




XI 


G 


CD 


x! g 


0) 




Q 


Q 




Eh 


< 


Q 


H 3 


jQ 


















03 




• 


. 




« 


, 


B 




H 




* — 


CM 




no 


=r 


LT> 


'a - ) 



58 



remaining four problems (3-6 in Table 4.1) comprise the major sources of 
discrepancy in chromatographic detection limits, and are the topics to 
be addressed in this report. 

Numerical Example 

Before proceeding, however, a numerical example which incorporates 
problems 4-6 (see Table 4.1) will be given. This example will 
demonstrate to the reader the magnitude of these problems and will 
facilitate later discussion. Though designed with liquid chromatography 
in mind, the points made by this example apply equally well to gas 
chromatography . 

The initial assumptions, experimental conditions, and results of 
this example are shown in Table 4.2. To avoid the confusion which it 
would certainly have caused, problem #3 of Table 4.1 was not 
incorporated into this example. If it had been, the results might have 
been even more shocking. Nevertheless, as seen in the bottom row of 
Table 4.2, the LODs for the two systems employing identical detectors 
differ by three and one-half orders of magnitude! 

Though a detailed explanation of this example is beyond the scope 
of the present discussion, the huge discrepancy in the two LODs will be 
reconciled after the last three problems in Table 4.2 are solved. This 
example should awaken the reader to the seriousness of these problems 
and demonstrate why they must be eliminated if the chromatographic LOD 
is to be a meaningful figure of merit. 






59 



Table 4.2. Example of widely differing detection limits 
Assumptions 

1, Liquid chromatograph with UV absorbance detector 

2. beer's Law applies, i.e., A = ebc. 



3. Analytical sensitivity, S = eb = 10,000 AU L mol 

4. Peak to peak noise, N D _ D = 2 x 10"^ AU 

5. Root mean square noise, N = 1/5 N 



-1 



Variables 


Experiment 


A 


Experiment ,B 


V inj 


5 uL 




20 uL 


LOD defn 


10 HL _/S 

P-P 




3 N /S 
- J rms 


V M (mL) 


2.5 




0.5 


k 


10 




3 


N (plates) 


1000 




10,000 



Results 

LODs reported 8.7 x 10" 6 M 

log (L0D A /L0D B ) = 3.5 orders of magnitude difference! 



3.0 x 10" 9 M 



60 

Solving the Problems 
Eliminating Mistaken Identities 

The limit of detection has unfortunately been confused with three 
other concepts—particularly the (minimum) detectability— which are also 
used in characterizing chromatographic trace analyses. Table 4.3 
includes symbols and definitions for all four of these concepts. 

One reason that the LOD is confused with the other concepts in 
Table 4.3, particularly the MD, is the redundancy in nomenclature of the 
detection limit and the MD, as evidenced by the partial, but 
representative list of (apparent) synonyms for these concepts (shown in 
Table 4.3) which appear frequently in the literature. In general, 
redundant terminology in science only serves to confuse. This is 
especially true when the apparent synonyms for different concepts are 
themselves quite similar. The use of these apparent synonyms should be 
discontinued immediately. 

Even if the confusion resulting from the redundant terminology 
could be eliminated, the LOD might still be confused with the MD by the 
apprentice chromatographer because their definitions, as shown in 
Table 4.3 and in eqs 4.1 and 4.2, are so similar in appearance (cf. 
meaning, however). 

LOD = arbitrary detector signal level/S (4.1) 

MD = arbitrary detector signal level/S d (4.2) 

£et despite their similarities, the LOD and MD are distinct 
concepts, as a closer scrutiny of Table 4.3 shows. The LOD is a general 
concept characterizing any overall trace analytical procedure consisting 
of one or more steps, whereas the MD is a specific term characterizing 



61 









-P G 






CO XI 






-P M 








x) 




•H CD 






-P 






•H G 








<r> 




G > 






•rH 






C -H 








-p 




3 -H 






X 3 






3 CO 








o 




bO 






to 






CO 








cd 










3 






CO 








+s 










O CD 






G O, 








a) 




G 






G CO 






CD 








XJ 


>» 


CD (0 






X -r-l 






a, 








G 


-P 


Q. 






-P O 






CD 








03 3 


•H 








c 






-p 








X XI 


> 


G 












-p >> 








a> 


•H 


-P -H 






bO CD 






3 rH 








c o 


-P 


3 






c x 






Q. cfl 








cfl O 


•r4 


a 






•H -P 






-P G 








o G 


CO 


-P T3 






co 




G 


3 CO 








a 


G 


3 CD 






co a 




O 


O 








-p 


CD 


O O 






CO O 




■H 


Cm 








«o <-i 


CO 


3 






Q. G 




■P 


O 








S3 CO 




•o 






tin 




cd 


rH 








-P O 


» 


rH O 






CD 




S-. 


CO r-V 








•rH 


•P 


CO G 






•P T3 




■p 


a cd 








cd .p 


C 


G -P 






>, CD 




G 


ho e 








-p >> 


3 


bO G 






rH G 




CD 


■H -r-i 






>) 


>>rH 





•H -H 






CO G 




V 


CO +> 






jB 


rH CO 


s 


CO 






C CD 




C 








o. 


CO C 


co 








CO O 




O 


1 -P 






cfl 


a co 




1 CD 






CO 




O 


•H 






G 


CO 


CD 


-P 






Cm -H 






CD G 






to 


c 


rH 


>> 






o -o 




CD 


> 3 




• 


o 


Cm CD 


X 


CD ^\ 










rH 


G \ 




>, 


-p 


O > 


CO 


> CO 






X CD 




X 


3 -P 




rH 


CO 


•H 


-P 


G G 






3 Xi 




CO 


CD G 




CD 


a 


-P bQ 


o 


3 CO 






rH 




-P 


3 




-P 


o 


G 


CD 


O 






<M a 




O 


CD O 




Cfl 

■H 


G 


3 CO 


J-5 








CO 




CD 


co e 




X 


O 


0) 


<w 






CO CD 




-P 


G CO 




X! 
CD 


O 


s g 


TD 


G O 






co 




CD 


o 






CO O 




O 






CO -p 




XI 


Q«-P 






c 


Ch 


a 


•iH 






a cfl 






CO -rH 




g 


■H 


G 


3 


-P -P 






X! 




a 


CD G 




•H 




O >> 


a 


CO G 






G -P 




3 


G 3 






CO 


•P 


•H 


G 3 






o 




e 






X! 


-P 


a c 


G 


x o 






CD 




•H 


G G 




CD 
3 


Q. 


O -H 


•H 


•H a 






c a 




c 


O CD 




CD 


■H CO 


a 


rH CO 






O -r-l 




•H 


-p a 




G 


O 


-p -p 




CO 


CD 




•iH JJ> 


>> 


8 


o 




■H 


c 


CO G 


•* 


o 


G 




•P 


-p 




CD 




4J 


o 


G CD 


>> 


G 


3 




CO -P 


G 


~ 


-P G 


G 


C3 


o 


■P o 


-p 


O 


T3 




G -H 


•H 


n-t 


CD O 


O 


o 




G 


•H 


CD 


CD 




•P G 


CO 


CD 


X> ^w' 


-P 


o 


co 


CD CD 


-P 


X 


O 




C 3 


-P 


> 




o 


CO 


■h 


O rH 


C 


•P 


O 




CD 


s- 


(D 


CD 


CD 


■rH 


co 


G X 


CO 


C 


G 




O 


CD 


rH 


XI G 


-P 


X3 


O CO 


3 


o 


Q 




c c 


O 




•P O 


CD 




H 


O G 


cr 


■rH 






O -H 




CD 


■iH 


"O 





to 


O 




Ch jJ> 


rH 




o 


CD 


rH 


HJ> 




J3 


C 


CO 


CI) 


O CO 


CO 






rH 


X 


Cm cfl 


CO 




cfl 


-P CO 


-P 


G 


O 




G 


X 


cd 


O G 




■a 




CO CD 


3 


•P 


■H 




a o 


CO 


-p 


-P 


X 


rH 


a> 


CD G 


rH 


c 


-P 




3 -P 


c 


o 


C 


M 


3 


o 


rH 


o 


CD CD 


>v 




e o 


O 


CD 


CD CD 


3 


o 


CO 


rH X 


CO 


Q, o 


H 




•H CD 


CO 


J-> 


a. O 


o 


XI 


G 


(0 -P 


£> 


O G 


CO 




G -P 


Cfl 


CD 


O G 


G 


CO 


H 


8 -H 


CO 


<-{ O 


G 




•rH CD 


CD 


X) 


r-i O 


G 






CO 3 




CO O 


CO 




a xt 


G 




CO O 


*J 


CO 




1 


a 
3 

a 


' 




>> 


i 




a 

3 

a 


J 




8 

g 


=t 




•H 






x 






•H 










c 






a. 






G 






a 


a> 


r™\ 


•H 


>, 




cfl 






•H 


,— «v 




>, 


<H 


a 


a 


-P 




G 






a 


X) 




CO 


X 


o 




•H 




bO 


>> 






CO 






cfl 


j 




> 




o 


+5 






'— * 




_p 


H 


•*s 


i 


-P 




-p 

CO 


•H 

rH 




i 


>, 




c 

CD 




G 




•H 




a 


•iH 






-p 




Q. 
CO 




O -P 


CO 


CO 




o 


XI 




cfl 


•H 






•H -H 


CO 


C 




G 


(0 




CO 


> 






•P S 


8 


(D 




X 


-P 




a 


•H 






O iH 


>> 


CO 




O 


o 




>> 


-P 






0) rH 


C 








CD 




c 


•rH 








-p 


o 


^*v 




O 


4J --> 




o 


CO 




CD 




CD G 


s 


<*-> &o 




-P 


CD Q 




G 


G 




CO 




X) O 


>^ 


rH -w 






x) S 




>, 


CD 




CD 




•iH 


CO 


C0 




o 






CO 


CO 




X 

^_3 




<m -P 




o 




■H 


,*-V. 












O O 


-P 


-H 




Cm 


a 




-p 


G 








CD 


G 


-P 




•rH 


3 




G 


o 




Cm 




•P 


CD 


>> 




O 


8 




CD 


-P 




o 




-P (D 


Li 


rH 




CD 


■H 




G 


o 








■H X) 


CO 


CO 




a. 


a 




CO 


CD 




CD 




a 


Cv 


G 




CO 


•rH 




D. 


-P 




CO 




•rH G 


O, 


CO 






3 




a. 


CD 




3 




rH O 


cd 






CO 

E 






Cfl 


XI 




CD 
X 




« 




e 




CD 


• 






« 




ca 




*™ 




C\J 




■P 


ro 






=r 











62 

one step in a chromatographic analysis: detection. For example, the 
LOD must, by definition, include the chromatographic dilution of the 
analyte, whereas the MD cannot. Furthermore, the LOD is measured 
experimentally with a complete chromatographic system (including the 
column) under the specific conditions of a given trace analysis; the MD, 
in contrast, is measured without a column under conditions that may or 
may not correspond to those of the analysis. 

The two concepts may also be distinguished mathematically. 
Assuming an analytical signal in terms of peak height, the relationship 
for a concentration sensitive chromatographic system is (36) 

LOD = (2tt) 1/2 [V M (l+k)/N 1/2 ] b MD (4.3) 

where V M represents the corrected gas holdup volume in GC and the column 
void volume in LC; k is the capacity factor; N is the number of 
theoretical plates; and b is a unitless parameter which permits the LOD 
and the MD to be defined independently of one another. For a mass 
sensitive chromatographic system, the relationship between the LOD and 
the MD is (37) 

LOD = (2tt) 1/2 [t M (1+k)/N 1/2 ] b MD (4.4) 

where t M is the retention time of an unretained solute, corrected for 
gas compressibility in GC. 

One final difference should be noted: An arbitrary detector signal 
level of twice the peak-to-peak detector noise has been universally 
agreed upon for MD calculations (in practice, at least). No such 
consensus exists for the LOD. 



63 

Choosing a Model 

As discussed earlier, the LOD may be defined in terms of an 
arbitrary signal-to-noise (S/l\l) level. In our survey of the 
chromatographic trace analysis literature, we found a multitude of 
arbitrary S/N levels used, ranging from 2 to 10. To further complicate 
matters, neither the measures of the signal (peak height, peak area, 
etc.) nor the noise (peak-to-peak, root mean square, etc.) were 
specified in many instances. 

These inconsistencies and ambiguities are not surprising since (to 
our knowledge) no standard model for the LOD has ever been proposed, 

much less adopted, by any recognized organization for the field of 

x 
chromatography! We note specifically the omission of an LOD definition 

in chromatography by the American Society for Testing and Materials 

(ASTM) and by IUPAC in their respective publications on gas and/or 

liquid chromatography nomenclature (38-41 ) . The omission by these and 

other organizations is also substantiated in two reviews (42,43). 

More importantly, however, the above inconsistencies and 

ambiguities can be eliminated completely if a clearly stated LOD model 

is adopted. Therefore the adoption of, with minor reinterpretation, the 

IUPAC model for spectrochemical analysis (44) or a model based on first 

order error propagation (10) is proposed. These models are given in 

eqs 4.5a and 4.5b, respectively, 

LOD = 3s B /S (4.5a) 

(The model which the International Union of Pure and Applied 
Chemistry (IUPAC) adopted in 1975 (44) was chosen specifically 
for spectrochemical analysis. Though the ACS Subcommittee on 
Environmental Chemistry reaffirmed this model in 1980 (45), 
it did so for the area of environmental chemistry and not 
specifically for the field of chromatography.) 



64 



LOD = 3[s B 2 + Si 2 + (i/S) 2 s s 2 ] 1/2 /S 



(4.5b) 



where S, i, s s , and s j _ are the analytical sensitivity (slope), 
intercept, and their respective standard deviations of the calibration 
curve obtained via linear regression; and s B is the standard deviation, 
calculated from 20 or more measurements of the blank signal. 

The factor of 3 in the numerator of the right hand expressions of 
eqs 4.5a and 4.5b gives a practical confidence level of 90$ to 99.7$, 
depending on the probability distribution of the blank signal and the 
accuracy of s B (10,44,46). Though smaller or larger factors could be 
used instead of 3, the resulting confidence levels would be too low or 
too high for practical use in most cases. Both the original proponents 
of these models (10,44) and others (46) strongly recommend the use of 
the factor 3. This author concurs. 

Both models are proposed for adoption because it seems preferable 
to let the chromatography community judge their respective merits. 
Indeed, strong arguments can be made for each. The IUPAC model, on the 
one hand, is computationally simpler and has already been employed, 
though infrequently, in the chromatographic literature. On the other 
hand, the error propagation model is not really all that complicated; 
many pocket calculators with linear regression capability can be easily 
programmed for the error propagation model. Furthermore, the error 
propagation LOD model takes uncertainties of the slope and intercept of 
the calibration curve into account, resulting in a more realistic 
numerical estimate. 

Interpretation . The IUPAC and error propagation LOD models were 
developed originally for spectroscopic trace analysis. Nearly all the 



65 

associated concepts, [e.g., the calibration curve, the sample (analyte + 

matrix), etc.] have identical, straightforward interpretations in 

chromatography. One aspect, however, does not: the measurement of s D . 

a 

Intuitively it is clear that the chromatographic baseline is 
somehow analogous to the blank signal in spectroscopy. We can now refer 
to s B as the standard deviation of the chromatographic baseline (noise). 
The measurement of Sg remains unclear, however. 

One possible procedure would be to estimate Sg from 20 or more 
measurements of only that portion of the baseline observed at the 
analyte' s retention time in the absence of the analyte (when a blank 
solution is injected). This is directly analogous to the measurement of 
the blank signal (at the analytical wavelength) in spectroscopy. Such a 
literal procedure would require at least 20 injections of blank solution 
(20 blank chromatograms!) and is obviously too impractical: 

1. It would be much too time consuming! 

2. It may require too much blank solution. 

3. Variables which affect retention would require strict control. 

4. The retention time of the analyte would need to be known very 
precisely. 

A much more practical procedure becomes apparent if one remembers 
that the standard deviation (root mean square) of a random (periodic) 
signal can be closely approximated by the quotient of the range (peak- 
to-peak displacement) and a parameter, p, dependent on the type of 
signal, i.e., 

Signal = ran §e/p (4.6a) 

For the measurement of s B , it should be noted that the range is 
equivalent to the peak-to-peak noise of the baseline, N , if the 
latter is measured over a sufficiently wide region of the chromatogram 
which includes the analyte peak. Additionally, the baseline usually 



66 



results from a normally distributed, random signal for which p = 5. 
Thus in most oases the standard deviation of the chromatographic 
baseline, Sg, can be estimated from one-fifth of the peak-to-peak noise, 
i.e. , 

S B = N p-P /5 (4 * 6b) 

Two additional comments regarding the practical procedure for 
measuring Sg should be noted: 

1. It is recommended that the "sufficiently wide region of the 
chroma togram" be at least as wide as 20 base widths of the analyte peak. 

2. If systematic fluctuations in the baseline are present, a value (of 
p) less than 5 should be used in eq 4.6b. If, for example, a periodic 
triangular baseline (possibly resulting from flow pulsations in the 
detector cell due to an insufficiently dampened solvent delivery system) 
is observed, then p = 3.5 and s B = N _ /3.5. 

Using the Correct Units 

A common misconception about chromatographic LODs is that they can 
be reported in units of concentration rather than amount. The fallacy 
of this assumption will be shown below. 

Dimensional analysis . One way of deducing the correct units of the 
chromatographic LOD is by dimensional analysis. Referring to the 
definition of the MD in Table 4.2, it is clear that if the appropriate 
units of concentration and mass flux are used for the MD in eqs 4.3 and 
4.4 for the concentration sensitive and mass sensitive detector cases, 
respectively, the units for the LOD in eqs 4.3 and 4.4 must be in terms 
of an amount (e.g., moles, grams, or some multiple thereof). 

Another approach via dimensional analysis is to consider the right 
hand expression of eq 4.1. Given the definition for the analytical 
sensitivity, it is clear that the units for this term (denominator of 






67 

eq 4.1) should be the quotient of the units of the measured signal and 
the units of the independent variable. Therefore, since the units for 
the noise expression (numerator of eq 4.1) are the same as those for the 
measured signal, the units for the detection limit should be the same as 
the units of the independent variable of the calibration curve. Thus, 
to decide which units are correct for the LOD, we need only to identify 
the units of the independent variable of the calibration curve, i.e., to 
determine whether the chromatographic signal depends on the 
concentration or amount of analyte injected. 

Equations 4.7 and 4.8 (36,37) below show that the signal (peak 
height, hp) is directly proportional to the maximum concentration, 

C max det' or ^ he max i- mum flux, F max <jet ' °^ ^ e G h romat °gr , aphic peak 
flowing through the detector, which in turn are directly proportional to 
the amount of analyte injected. 

h P - C max,det = ^inj » V2 C**> 1/2 /Vr (4.7) 

h P * F max,det = ^inj ""* ^ U2 /H (4.8) 

Thus the same conclusion is reached once again: The chromatographic LOD 
must be given in units of amount, i.e., in moles, grams, or some 
fraction thereof. 

Identifying faulty logic . Despite these convincing arguments, some 
researchers insist on reporting their chromatographic LODs in 
concentration units. Their rationale might be as follows: 

1. The amount of analyte injected, Q.± n i' is the product of the 
concentration of analyte in the sample, C, ,, and the volume of sample 
injected, ^ . . 

«inj = C inj V inj <*•» 






68 

2. From eq 4.9 it is clear that the concentration of analyte injected, 
C inj' is direc tly proportional to the amount of analyte injected. 

C inj = W V inj (4 ' 10) 

3. Therefore the relative LOD (in units of concentration), C,, is 
proportional to the true, absolute LOD, q L (in units of amount). 

C L = 0L /V inj (*.") 

Though reached in a straightforward manner, the conclusion stated 

above is nevertheless false. The error in reasoning is best described 

as an improper or incomplete analogy. In going from a true expression, 

eq 4.10, to a false statement, eq 4.11, C. ■ and q. . were replaced by 

mj inj r J 

two limiting quantities C L and q L , respectively. No analogous 
substitution was made for V in , , however, and therein lies the error. 
V inj may vary continuously over < V inj < V inj>max , where V inj|fflax is 
some limiting, maximum injection volume to be discussed momentarily. 
Unless V. nj = V inj)fflax , eq 4.11 is false. 

A numerical example will help demonstrate the absurdity of eq 4.11. 
Suppose the true, absolute LOD (q L ) for analyte X had 'been determined 
independently by two scientists using the same LC system to be 1 x 10~ 12 
mol. If the scientists had used different injection volumes of 5 uL and 
50 uL, according to eq 4.11 the relative LODs (C L 's) for the same 
chromatographic system would be 2 x 10" 6 M and 2 x 10" 7 M, respectively. 
Clearly eq 4.11 is inappropriate. 

The correct expression, eq 4.12 below, is obtained by using 
V inj,max in Place of V in j . But this expression is of little value 
because of the difficulty in obtaining a consistent, precise estimate of 

V. 

in j ,max" 

C L = «L /V inj,max < 4 ' 12 > 



69 



Problems with estimating the maximum injection volume . Experimen- 
tally, Vj_ n -j max °an be determined by increasing V. ., until column over- 
loading or some other adverse phenomenon is observed. This operational 
definition of V in - max is unsatisfactory, however, because the injection 
volume at which these events occur is too dependent on the experimental 
conditions, e.g., the sample matrix, the percent loading of the column. 

Alternatively, numerous theoretical expressions are available for 
the estimation of V in - max . Though many are overly specific, a few are 
completely general. Perhaps the best is one which relates V- ■ to 
the maximum tolerable degradation in resolution (5, p. 289). Our 
extension of this expression is reported below as eq 4.13, and is 
derived in Appendix C. 

V inj,max = 2 (3) 1/2 (y 2 + 2y) 1/2 V M (1 + k)/N 1/2 (4.13a) 

where y = the maximum tolerable loss in resolution due to a finite 
injection volume. 

Another expression based on the same criterion (maximum tolerable 
loss in resolution) but derived from slightly different assumptions (36) 
is shown in eq 4.13b 

V inj,max = < 2K) * V M (1 + *)/N 1/2 (4.13b) 

where K is a parameter characteristic of the method of injection; 
practical values of K range from 2 to 3.5 (ideal plug injection). 

Comparison of eqs 4.13a and 4.13b reveals one of the problems with 
theoretical estimates of V^ . max : due to their extreme dependency on 
the criterion selected and/or the assumptions of the derivation, 
theoretical estimates of V i . max are extremely inconsistent. 



70 

Another problem with these and other general expressions is that 

V inj max is ni S nl y dependent on the value of some arbitrary parameter, 

y in this example (eqs 4.13a and 4.13b). If y = 0.04 (k% loss in 

resolution), eq 4.13a becomes 

V- = V, .( 1 + k)/N 1/2 (4 14) 

v mj,max >r ' + K;/w ^ ,l4 ' 

If y were chosen to be 0.01 or 0.10, then V . . _„„ would be equal to 

LI I J j ludX 

approximately half or one and one-half times the value predicted by 
eq 4.14. Since universal agreement on the value of y or some other 
parameter is unlikely, the theoretical methods for estimating V . . mov 

XII J y ul3-X 

will continue to be imprecise. 

Summary . For all the reasons discussed above, chromatographic LODs 
should always be reported in units of amount, not concentration. 

Converting to Chromatographic Reference Conditions 

As seen from eqs 4.3 and 4.4, the LOD depends not only on the 
detector characteristics (MD) but also on three chromatographic 
parameters [k, N, and V^ (or t M )] which characterize the column and the 
solute. In this section a method is proposed for taking the effects of 
these parameters into account, thereby solving the final problem 
associated with the chromatographic LOD (see Table 4.1). Gaussian 
elution profiles are assumed, though slight to moderate deviations can 
be tolerated. 

B'or a concentration sensitive detector, the relationship between 
the LOD and the chromatographic parameters is given by 

q L = d V M (1+k)/N 1/2 (4.15) 

where terms of eq 4.3 [(2ir) 1/2 , b, MD] have been incorporated into a 






71 



single proportionality constant, d. Alternatively, since V„ = V.,(1 + k) 



R " "M v 



and N = (V^c^) 2 , we may write 



q L = d V R /N 1/2 (4.16) 

or q L = d a y (4.17) 

where V R is the retention volume (corrected for compressibility in GC) 
and cfy is the bandwidth of the chromatographic peak, in volume units. 

Equations 4.15-4.17 are equivalent, and any one of them may be used 
to derive an expression which accounts for the effects of these 
chromatographic parameters on the LOD. For simplicity, we use eq 4.17. 

Consider a LOD, q L1 , obtained under one set of chromatographic 
conditions, i.e., o y = a v1< By analogy with eq 4.17, 

Q L 1 = da V1 ( 4 - 18 ) 

Likewise, for a LOD, q L2 , obtained under a second set of chromatographic 

conditions, 

%2 = da V2 (4.19) 

Dividing eq 4.19 by eq 4.18 and solving for q L2 yields 

^L2 = ^ G V2 /a V1 ] QL1 (4.20a) 

which can also be written as 

q L2 = CV R2 /V R1 ] tN 1 /N 2 ] 1/2 q L1 (4.20b) 

or q L2 . LV M2 /V M1 ] [(k 2 + 1)/(k 1 + 1)] [N^Ng] 172 q L1 (4.20c) 

Equations 4.20a-4.20c are important for two reasons: First, they 
can be used to predict the change in the detection limit when switching 



72 



from one set of chromatographic conditions to another. Thus they are 
useful to the analyst interested in lowering the detection limits via 
improvements in the chromatographic (rather than the detection) aspects 
of the trace analysis. It should be noted, however, that the detection 
limits cannot be lowered infinitely by such improvements. Eventually 
the analysis will be optimized to the point where further improvements 
in the chromatographic aspects will necessitate a reduction in sample 
volume which offsets the chromatographic improvements (36). Of course, 
in situations where very little sample is available, this reduction in 
sample size while maintaining a constant detection limit will be 
helpful. 

Second, eqs 4.20a-4.20c can be used to compare detection limits 
obtained under different chromatographic conditions (bandwidths) . 
Consider two such detection limits obtained using different 
concentration sensitive detectors. These detection limits can be 
compared only after one of the two detection limits is converted from 
its present value obtained under a certain chromatographic bandwidth 
(set of conditions) to a value corresponding to the bandwidth of the 
other LODj, or after both LODs are converted to values corresponding to a 
third bandwidth. Note that the latter would require an additional 
conversion. 

Whichever way is chosen, it should be recognized that the final 
bandwidth (set of chromatographic conditions) chosen in the LOD 
conversion procedure serves as a reference state and that the initial 
bandwidth(s) is(are) by definition experimental states. Thus from 
eq 4.20a we may write 

<*L,ref = £a v>ref /a Vjexp ] q L ,exp ( 4 - 21 > 



73 



where the subscripts "ref" and "exp" refer to reference and 
experimental, respectively. Similar equations resulting from the 
incorporation of this notation into eqs 4.20b and 4.20c are easily 
inferred. 

The above derivation can be repeated in an analogous fashion for 
the case of mass sensitive detectors. The result is 

*L,ref = [a t,ref /a t,exp ] ^L,ex P " -.22) 

where a t is the bandwidth of the peak in time units. 

Given eqs 4.21 and 4.22, the analyst now has a method for comparing 
detection limits obtained under different chromatographic conditions 
(states). Given two or more LODs measured under different 
chromatographic states, the analyst merely converts all of them to 
values corresponding to a single, arbitrary, reference state. 

Though the choice of the reference state is arbitrary, the 
selection of the reference state is nevertheless important. For 
example, it would be counterproductive for an analyst to employ a 
different chromatographic reference state each time a new group of 
experimental LODs (measured under different experimental states) were to 
be compared, since LODs in separate groups could not be compared if this 
was done. For similar reasons, it would also be counterproductive if 
each analyst or group of analysts used different reference states. 

On the other hand, the reference states by definition must be 
different for concentration sensitive and mass sensitive chromatographic 
systems. And in addition, typical chromatographic states (bandwidths) 
vary considerably in liquid chromatography (LC), packed column gas 
chromatography (PGC), and open tubular gas chromatography (OTGC). 



74 

The logical compromise which is proposed is a fixed reference state 
(bandwidth) to be used exclusively for each of the three chromatographic 
areas (LC, PGC, and OTGC) in each of the two detector categories 
[concentration sensitive (eq 4.20) or mass sensitive (eq 4.21)], giving 
a total of six fixed references states. Furthermore, it is proposed 
that the LODs resulting from the conversion to these fixed reference 
states be referred to as standardized chromatographic LODs. To 
emphasize this, eqs 4.21 and 4.22 may be rewritten as 

a L,std = [G V,ref /G V,exp ] a L,exp (4 ' 2 3) 

and q L,std = K,ref /a t,expJ ^L,exp ^- 24 ) 

respectively, where the standardized LODs are indicated by the subscript 
"std". 

Advantages of the chromatographic reference state concept and the 
resulting standardized chromatographic LODs . If the values for the 
reference states are chosen judiciously, the resulting standardized LODs 
will be superior to experimental (hereafter termed conventional ) LODs in 
several ways: 

First, standardized LODs generally represent a more realistic 
measure of the trace analysis capabilities of a given chromatographic 
system, thus permitting an analyst to decide whether or not a particular 
application reported in the literature will be feasible with his or her 
system. Conventional LODs, on the other hand, may be very misleading. 
As evidenced in the literature, many trace analyses have been performed 
using chromatographic systems which have only been partially optimized, 
if at all. As a result, unusually pessimistic (very high) LODs are 
obtained. In contrast, some overly optimistic (very low) LODs have been 



75 



reported in some instances where the particular chromatographic systems 
have been optimized more than the typical system could have been. 

Second, the use of standardized LODs permits a fair comparison of 
trace analysis chromatographic systems in different areas, e.g., OTGC 
vs. LC, or with different types of detectors (mass vs. concentration 
sensitive). Such a comparison is not valid if conventional LODs are 
used, even within a given area using the same type of detector. 

Third, the standardized LOD facilitates the prediction of the value 
of the LOD under experimental chromatographic conditions. Rearranging 
eqs 4.23 and 4.24, one obtains 

%,exp = [a V,exp /a V,ref ] ^L,std ( 4 -25) 

&nd q L,exp = [G t,exp /a t,refJ ^L,std ^.26) 

Assuming q LjStd has been reported, one merely needs to substitute values 
for a V,exp (or Vexp) in e 3 4 -25 °r *• 26 in order to calculate q r 

Fourth, the standardized LOD can serve not only as a figure of 
merit for the overall chromatographic analysis, but as a figure of merit 
for the detector as well. The need for the minimum detectability (MD) 
as a separate parameter (and hence, the need for a detector calibration 
curve) would therefore be eliminated, resulting in a considerable 
savings of time. 

Numerically defining the refere nce states . Values for the proposed 
chromatographic reference states (bandwidths) were determined by 
assigning values to the component parameters [V [v] (or t M ) , k and N] and 
then calculating the reference bandwidths using eq 4.27 or 4.28: 

a V,ref = V M,ref (1 + k ref )/(N ref )1/2 ^- 2 7) 



76 



CT t,ref = ^.ref + k ref )/(N ref )1/2 (4.28) 

The reference values for the individual parameters were selected in 
accordance with one or more of the following criteria: 

1. The value represents an intermediate chromatographic performance 
which is easily achieved except under unusual circumstances. 

2. The value falls within a range of values reported directly in the 
scientific and trade literature. 

3. The value falls within a range of values calculated from column 
manufacturer specifications. 

4. The value falls within a previously recommended range [k only— see 
ref 12, p. 67]. 

The values selected for the chromatographic bandwidth component 
parameters are given in Table 4.4. The resulting reference states and 
corresponding standardized LOD equations are shown in Table 4.5. 

Table 4.5 and the advantages enumerated earlier conveniently 
summarize the bulk of the material presented in this section. Only one 
additional point needs to be made: non-standard (experimental) 
bandwidths are best measured directly using equations such as 

G = V 4 (4.29) 

° r a G = W 0<1 /4.292 (4 >30) 

where W fa and W Q#1 represent the base width and the width at 10% peak 
height, in units of volume or time, whichever is appropriate. 
Alternatively, the bandwidths can be calculated from their component 
parameters, but this is disadvantageous in two respects. First, it 
requires at least two measurements instead of one. Second, different 
expressions and/or different interpretations are required for gradient 
(temperature or mobile phase) elution conditions. 



77 



Table 4.4. Proposed values for the component parameters 
of the chromatographic reference states 



parameter LC PGC OTGC 



k ref 



N ref 10,000 



V M>ref (mL) 



^.ref^ 1 "') 1 



10,000 


62,500 


3 


2 


0.3 


1.3 



78 



Table 4.5. Proposed standardized LOD equations and 
the corresponding chromatographic reference states 



q L,std = tCT V,ref /a V,exp ] q L,exp concentration 

sensitive detector 



q L,std = ta t,ref /CT t,exp ] q L,exp mass sensitive 









detector 




LC 


PGC 




OTGC 


°V,ref (mL) 


0.05 


0.15 




0.04 


^.ref^ 111 -) 


0.05 


0.015 




0.026 



79 



The Numerical Example - Revisited 

We return to our hypothetical LOD example (Table 4.2 and associated 
text) to reconcile the large differences in the reported detection 
limits. Recall that since identical chromatographic detectors were 
employed, the huge discrepancies were attributed solely to problems 4-6 
of Table 4.1. By eliminating these problems one at a time, obtaining 
identical detection limits at the conclusion, this claim will now be 
proven. 

The reconciliation is summarized in Table 4.6, though readers who 
wish to perform the calculations will need to refer to conditions 
specified in Table 4.2. 

The progress of the LOD reconciliation may be noted by inspecting 
either the LOD values themselves in the second and third columns, or 
their ratio given in the fourth column, in orders of magnitude. The 
degrees to which the given problems are responsible for the initial 
discrepancy between the LODs are shown in the fifth column; they are 
obtained by subtracting successive values in the adjacent column. 

Three steps were performed in the reconciliation. First, the LOD 
values reported incorrectly in units of concentration (mol L" 1 ) are 
converted to the appropriate units of amount (mol) by multiplying by the 
corresponding sample injection volumes (V in j's). Second, these unit- 
corrected LODs are then converted to values consistent with the IUPAC 
model previously discussed (q L = 3 s B /S). For case A, since N _ = 5s R 
(as discussed earlier in Choosing a Model ), the LOD must be reduced by a 
factor of 50/3. For case B, since N rms = s B , no adjustment is 
necessary. Finally, the IUPAC consistent LOD values are converted to 






80 



Table 4.6. Reconciling the differences in the detection limits 
from the numerical example in Table 4.2 



Step 



LOD, 



L0D t 



log 



\ L0 V 



log 



0. Initial 



1 . Amount 



8.7 x 10 -6 H 3.0 x 10~ 9 M 



44 pmol 



2. IUPAC def'n 2.6 pmol 



3. q L)Std 150 fmol 



60 fmol 



60 fmol 



150 fmol 



3.5 



2.9 



1.6 



0.0 



0.6 

1.3 
1.6 



81 

standardized LODs, thereby adjusting for differences in the 
experimental chromatographic conditions (bandwidths, or states). 

As 3een in Table 4.6, the discrepancy between the LODs is reduced 
significantly with every successive stage. It should be noted that the 
largest source of discrepancy is due to differences in the experimental 
chromatographic conditions (bandwidths). This demonstrates the need for 
a standardized chromatographic limit of detection. Finally, the LODs in 
the bottom row of Table 4.6 are identical, indicating that the 
reconciliation has been completely successful. 

Conclusion 
Meaningful chromatographic detection limits can be obtained only if 
careful attention is paid to the application of the principles which 
have been discussed: 

1. The limit of detection (LOD) should not be confused with the 
(analytical) sensitivity (S), the minimum detectability (MD), or the 
detector sensitivity (S,). 

2. The experimentally observed LODs should be calculated using the IUPAC 
and/or the error propagation model(s). The calibration curve should be 
constructed from a plot of signal versus amount (not concentration!) of 
analyte injected, thus insuring that the resulting LODs will be in units 
of amount (not concentration!). 

3. If obtained under non-standard conditions, the detection limits can 
be standardized using the equations in Table 4.5. Standardized 
chromatographic LODs are superior to their conventional (non- 
standardized) counterparts for several reasons, in particular because 
they permit the (valid) comparison of trace analysis chromatographic 
systems in different areas (LC, PGC, and OTGC) and/or with different 
types of detectors (mass and concentration sensitive). 



CHAPTER 5 
SUGGESTIONS FOR FUTURE WORK 



The results of this work should facilitate and encourage the study 
of a number of related and unrelated topics. A few examples are given 
below. 

With regard to the standardized chromatographic LOD concept 
introduced in Chapter 4, if open tubular liquid chromatograpy ever comes 
of age, it will be desirable to extend the LOD concept to this area by 
defining two additional reference states (bandwidths) . 

The successful generation of the exponentially modified Gaussian 
(EMG) function by microcomputer and subsequent measurement of the 
associated universal data should encourage others to use this realistic 
model in related or unrelated modeling studies. One project which has 
already been suggested by Kirkland et al. (23) is the development of a 
chromatographic resolution function (equation) which is applicable to 
skewed peaks as well as to ideal ones. 

The equations recommended in Chapter 3 for the characterization of 
skewed (EMG) and Gaussian peaks could be tested on analomous peaks 
resulting from column overloading, nonlinear distribution isotherms, and 
other sources of analomous peak distortion. 

Since the graphical deconvolution of the symmetric (^q 2 ) and 
asymmetric (t ) sources of band broadening is now possible via eqs 3 and 
4 in Table 3.2, the next step would be the development of methods for 
measuring the individual components of ov. and i . 

82 



83 



And finally, the relative error (fi£) approach used so successfully 
in the derivation of the equations in Chapter 3 is completely general 
and should be considered by other scientists for any semi-empirical 
modeling in their fields. 






APPENDIX A— DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMG-U 

EMG-U, the BASIC program which evaluates the EMG function and 
obtains the universal data, is listed below. In addition to the 
documentation supplied within the program itself and the flowcharts 
given in Fig 2.1, the following should be noted: 

1. By letting S = a Q / T and Y = (t-t Q )/a G , eq 2.4 can be written as 

h EMQ (t) = [AS(2tt) 1/2 ] exp(S 2 /2 - SY) /exp(-y 2 /2)dy/(2T:) 1/2 ( A .1) 

J -co 

where z = Y - S. (A ?) 

2. To the extent allowed by BASIC, the symbolism used in EMG-U is 
consistent with that in eqs A.1 and A. 2 and elsewhere in the text. 

Greek symbols were spelled out partially or entirely. 

3. EMG-U was developed on an Apple II Plus computer and optimized for 
0.1 < t/o g < 4 using t Q = 100, a Q = 5, 0.001 < dt fflin < 1, and 0.001 < a 
< 1 . Minor modifications may be required for optimum performance if 
other computers or other values for the parameters are used. 

4. EMG-U was designed to minimize execution time; major reductions are 
not likely to be achieved unless a compiler is used. 

5. The EMG evaluation subroutine (lines 2990-3160) may be used 
independently of the other routines in EMG-U. 



84 



85 

100 REM EMG-U EVALUATES THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) 

FUNCTION 
110 REM ALSO DETERMINES TR, HP, TA, TB, B/A, W/SIG, AND (TR-TG)/SIG 
120 B$ = •» ADD LINE»:C$ = » 3150 PRINT T,HEMG":D$ = "TO HELP DEBUG." 

130 ZERO = 0:PT5 = .5:WUN = 1:TWO = 2:RTTWOPI = SQR (TWO * 
3. 141592654): TEN = 10: K4 = 10000:K5 = 100000: REM COMMON NUMERICAL 
CONSTANTS 

140 P = ,2316419:B1 = .31938153:B2 = - ,356563782:B3 = 1 . 78 1 477937 : B4 
= - 1.821 255978 :B5 = 1.330274429: REM CONSTANTS FOR 14 POLYNOMIAL 
APPROX. 

150 A = 1:TG = 100:SIG = 5: REM EMG PRE-EXPONENTIAL CONSTANTS 

200 INPUT "ALPHA = (.005 < ALPHA < 1) ? »; ALPHA 

220 PRINT "INITIAL, FINAL VALUES OF TAU/SIGMA (T/S)»: INPUT "(0 1 <- 

T/S <= 4.25) ? ";R1,R2 
240 INPUT "TAU/SIGMA INCREMENT ? ";RSIZE 
260 INPUT "DT MIN = ? ( .001 , .01 , . 1 , 1 S) »;TLR 

300 REM BEGIN DISK STORAGE OF EMG DATA (OPTIONAL) 

400 FOR R = R1 TO R2 STEP RSIZE 
440 T = R: GOSUB 3300:R = T:S = WUN / R 
480 GOSUB 1000: REM GO TO MAIN EMG DATA ROUTINE 

500 BA = (TB - TR) / (TR - XTA):WS1G = (TB - XTA) / SIG:RR = (TR - TG) 
/ SIG 

540 BA = INT (BA * K4 + PT5) / INT (K4 + PT5):T = RR: GOSUB 3300 :Rfl 

600 PRINT R,BA: PRINT WSIG,RR: PRINT 

700 TR = ZERO: HP = ZERO: XTA = ZEROrTB = ZERO 

720 T1 = ZER0:T2 = ZERO:DT = ZERO:HTEMP = ZERO 

800 NEXT R 

830 REM END OPTIONAL DISK STORAGE OF EMG DATA 
850 END : REM PROGRAM HAS RUN TO COMPLETION 

1000 T = TG: GOSUB 3000:HTEMP = HEMG:DT = WUN 

1020 T s T + DT: GOSUB 3000 

1040 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: GOTO 1100 

1060 HTEMP = HEMG: GOTO 1020 

1080 GOSUB 3350 

1100 IF DT < = TLR THEN 1160 

1120 GOSUB 3450: GOSUB 1620 

1140 GOTO 1100 

1160 TR = T2 - DT:T = TR: GOSUB 3000: HP = HEMG:HLOOK = HP * ALPHA 

1200 T = INT (TR + WUN + PT5) 

1220 T = T - WUN: GOSUB 3000 

1240 IF HEMG > HLOOK THEN 1220 

1260 DT = WUN: GOSUB 3400 

1280 IF DT < = TLR THEN 1340 

1300 GOSUB 3450: GOSUB 1830 

1320 GOTO 1280 

1340 GOSUB 3500: XTA = T: GOSUB 3000 






86 



1400 T = INT (TR + PT5) - INT (SIG + PT5) 

1420 T = T + INT (SIG + PT5): GOSUB 3000 

1440 IF HEMG > HLOOK THEN 1420 

1460 TUT- INT (SIG + PT5):T2 = T:DT = WUN: GOSUB 2030 

1460 IF DT < = TLfi THEN 1540 

1500 GOSUB 3450: GOSUB 2030 

1520 GOTO 1460 

1540 GOSUB 3500.-TB = T: GOSUB 3000 

1560 RETURN : REM END OF MAIN EMG DATA ROUTINE 

1600 REM TR SEARCH LOOP BELOW THRU LINE 1720 

1620 FOR T = T1 TO T2 + DT STEP DT 

1640 GOSUB 3000 

1660 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: RETURN 

1660 HTEMP = HEMG 

1700 NEXT T 

1720 A$ = "THE TR SEARCH HAS FAILED!": GOSUB 3550: END 

1800 REM TA SEARCH LOOP BELOW THRU LINE 1950 

1830 FOR T = T2 TO T1 - DT STEP - DT 

1860 GOSUB 3000 

1890 IF HEMG < HLOOK THEN GOSUB 3300: GOSUB 3400: RETURN 

1920 NEXT T 

1950 A$ = "THE TA SEARCH HAS FAILED!": GOSUB 3550: END 

2000 REM TB SEARCH LOOP BELOW THRU LINE 2150 

2030 FOR T = T1 TO T2 + DT STEP DT 

2060 GOSUB 3000 

2090 IF HEMG < HLOOK THEN GOSUB 3300 :T = T - DT: GOSUB 3400: RETURN 

2120 NEXT T 

2150 A$ = "THE TB SEARCH HAS FAILED!": GOSUB 3550: END 

2990 REM EMG EVALUATION SUBROUTINE BELOW THRU LINE 31 60 

3000 I = (T - TG) / SIG:E = S * S / TWO - S * Y:Z = Y - SiZTEMP = ZERO 

3020 IF Z > = ZERO THEN 3060 

3040 ZTEMP = Z:Z = ABS (Z) 

3060 NF = EXP ( - Z * Z / TWO) / RTTWOPI:Q = WUN / (WUN + P * Z) 

3080 PQ = * (B1 + Q * (B2 + Q * (B3 + Q * B4 + B5 * Q * Q))) 

3100 I = NF * PQ 

3120 IF ZTEMP > = ZERO THEN I = WUN - I 

3140 HEMG = A * S * RTTWOPI * EXP (E) * I 

3160 RETURN 

3300 T = INT (T * K5 + PT5) / INT (K5 + PT5): RETURN 

3350 T1 = T - TWO * DT:T2 = T: HTEMP = ZERO: RETURN 

3400 T1 = T:T2 = T + DT: RETURN 

3450 DT = DT / TEN: RETURN 

3500 T = (T1 + T2) / TWO: GOSUB 3300: RETURN 

3550 PRINT CHR$ (7): PRINT A$;B$: PRINT C$: PRINT D$: RETURN 



APPENDIX B— UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, 0.50 



T/a G 

0.3 

0.35 

0.4 

0.45 

0.5 

0.55 

0.6 

0.65 

0.7 

0.75 

0.8 

0.85 

0.9 

0.95 

1.0 

1.05 

1.1 

1.15 

1.2 

1.25 

1.3 

1.35 

1.4 

1.45 

1.5 

1.55 

1.6 

1.65 

1.7 

1.75 

1.8 

1.85 

1.9 

1.95 

2.0 



0.05 



(B/A >0.05 


W 0.05 /CT G 


(t R -t Q )/a G 


1.0331 


5.1088 


0.2792 


1.0490 


5.1812 


0.3194 


1.0679 


5.2620 


0.3576 


1.0898 


5.3506 


0.3938 


1.1141 


5.4468 


0.4280 


1.1411 


5.5494 


0.4608 


1.1701 


5.6582 


0.4920 


1.2009 


5.7722 


0.5218 


1.2337 


5.8912 


0.5502 


1.2679 


6.0146 


0.5774 


1.3037 


6.1416 


0.6034 


1.3404 


6.2722 


0.6284 


1.3783 


6.4056 


0.6522 


1.4170 


6.5414 


0.6752 


1.4563 


6.6796 


0.6974 


1.4964 


6.8196 


0.7186 


1.5368 


6.9610 


0.7392 


1.5776 


7.1040 


0.7590 


1.6189 


7.2482 


0.7782 


1.6605 


7.3934 


0.7966 


1.7022 


7.5392 


0.8146 


1.7441 


7.6862 


0.8320 


1.7862 


7.8336 


0.8488 


1.8285 


7.9816 


0.8650 


1.8706 


8.1302 


0.8610 


1.9132 


8.2792 


0.8964 


1.9555 


8.4286 


0.9114 


1.9982 


8.5784 


0.9258 


2.0407 


6.7286 


0.9400 


2.0833 


8.8788 


0.9538 


2.1259 


9.0296 


0.9672 


2.1685 


9.1804 


0.9604 


2.2111 


9.3314 


0.9932 


2.2535 


9.4826 


1.0058 


2.2961 


9.6338 


1.0180 



87 



T/a G ""^0.05 W 0.05 /a G (t R -t )/a G 

2 - 05 2.3367 9.7856 1.0298 

2-1 2.3813 9.9370 1.0414 

2 - 1 5 2.4237 10.0890 1.0528 



2.2 

2.25 

2.3 

2.35 

2.4 

2.45 

2.5 



2.65 



2.75 

2.8 

2.85 



2.95 



(B/A) 0>05 


W 0.05 /a G 


2.3387 


9.7856 


2.3813 


9.9370 


2.4237 


10.0890 


2.4662 


10.2406 


2.5084 


10.3926 


2.5509 


10.5448 


2.5934 


10.6968 


2.6355 


10.8490 


2.6778 


11.0010 


2.7200 


11.1534 


2.7620 


11.3056 


2.8044 


11.4580 


2.8465 


11.6104 


2.8885 


11.7628 


2.9303 


11.9152 


2.9723 


12.0678 


3.0143 


12.2202 


3.0561 


12.3726 


3.0978 


12.5252 


3.1395 


12.6776 



1.0640 
1.0750 
1.0856 
1.0960 
1.1064 
1.1164 
1.1262 



2 ' 55 2.7620 11.3056 1.1360 

2.6 



1.1454 
1.1546 



2-7 2.8885 11.7628 1.1638 



1.1728 

1.1816 

1.1902 

2 -9 3.0561 12.3726 1.1988 



1.2072 



3-0 3.1395 12.6776 1.2154 






89 



Va Q ( B /A) 0<1 

0.1 1.0013 

0.15 1.0043 

0.2 1.0096 

0.25 1.0174 

0.3 1.0278 

0.35 1.0406 

0.4 1.05&0 

0.45 1.0734 

0.5 1.0927 

0.55 1.1140 

0.6 1.1367 

0.65 1.1608 

0.7 1.1864 

0.75 1.2133 

0.8 1.2412 

0.85 1.2700 

0.9 1.3001 

0.95 1.3309 

1.0 1.3621 

1.05 1.3942 

1.1 1.4269 
1.15 1.4599 

1.2 1.4935 
1.25 1.5274 

1.3 1.5615 
1.35 1.5960 

1.4 1.6307 
1.45 1.6658 

1.5 1.7006 

1.55 1.7358 

1.6 1.7712 
1.65 1.8066 

1.7 1.8419 
1.75 1.8775 

1.8 1.9128 
1.85 1.9466 

1.9 1.9843 
1.95 2.0198 
2.0 2.0555 



= 0.10 



W 0.1 /G G 


(t R -t G )/a Q 


4.3142 


0.0940 


4.3412 


0.1440 


4.3764 


0.1916 


4.4200 


0.2366 


4.4710 


0.2792 


4.5292 


0.3194 


4.5932 


0.3576 


4.6628 


0.3938 


4.7372 


0.4282 


4.8158 


0.4608 


4.8984 


0.4920 


4.9844 


0.5218 


5.0736 


0.5502 


5.1660 


0.5774 


5.2608 


0.6034 


5.3584 


0.6284 


5.4580 


0.6522 


5.5598 


0.6752 


5.6632 


0.6974 


5.7684 


0.7186 


5.8748 


0.7392 


5.9S28 


0.7590 


6.0918 


0.7782 


6.2016 


0.7966 


6.3124 


0.8146 


6.4242 


0.8320 


6.5364 


0.8486 


6.6492 


0.8659 


6.7626 


0.8810 


6.8764 


0.8964 


6.9908 


0.9112 


7.1054 


0.9258 


7.2204 


0.9400 


7.3356 


0.9538 


7.4512 


0.9672 


7.5670 


0.9804 


7.6830 


0.9932 


7.7992 


1.0058 


7.9154 


1.0180 






90 



Va G 

2.05 

2.1 

2.15 

2.2 

2.25 

2.3 

2.35 

2.4 

2.45 

2.5 

2.55 

2.6 

2.65 

2.7 

2.75 

2.6 

2.85 

2.9 

2.95 

3.0 



;b/aj 0#1 


W 0.1 /a G 


(t fi -t G )/a G 


2.0913 


8.0320 


1.0298 


2.1270 


8.1486 


1.0414 


2.1627 


8.2652 


1.0528 


2.1985 


8.3822 


1.0640 


2.2340 


8.4992 


1.0750 


2.2697 


8.6160 


1.0856 


2.3054 


8.7332 


1.0960 


2.3409 


8.8504 


1.1064 


2.3765 


8.9676 


1.1160 


2.4122 


9.0850 


1.1262 


2.4475 


9.2024 


1.1360 


2.4831 


9.3198 


1.1454 


2.5184 


9.4374 


1.1548 


2.5541 


9.5546 


1.1638 


2.5893 


9.6722 


1.1728 


2.6249 


9.7898 


1.1816 


2.6603 


9.9074 


1.1902 


2.6955 


10.0248 


1.1988 


2.7307 


10.1426 


1.2072 


2.7659 


10.2602 


1.2154 






91 



0.30 



x/a G 

0.3 

0.35 

0.4 

0.45 

0.5 

0.55 

0.6 

0.65 

0.7 

0.75 

0.8 

0.85 

0.9 

0.95 

1.0 

1.05 

1.1 

1.15 

1.2 

1.25 

1.3 

1.35 

1.4 

1.45 

1.5 

1.55 

1.6 

1.65 

1.7 

1.75 

1.8 

1.85 

1.9 

1.95 

2.0 



(B/A) 



0.3 



1.0185 
1.026b 
1.0366 
1.0474 
1.0592 

1.0722 
1.0857 
1.0998 



,1149 

,1303 

,1463 

1627 

1800 

1973 

1.2151 

1.2334 
1.2520 
1.2709 
1.2900 
1.3095 
1.3293 
1.3493 
1.3696 
1.3902 
1.4108 

1.4317 
1.4527 
1.4742 
1.4955 
1.5170 
1.5387 
1.5606 
1.5825 
1.6043 
1.6263 



W 0.3 /a G 


(t R -t G )/a G 


3.2248 


0.2792 


3.2616 


0.3194 


3.3014 


0.3576 


3.3434 


0.3938 


3.3878 


0.4282 


3.4336 


0.4608 


3.4806 


0.4920 


3.5290 


0.5218 


3.5784 


0.5502 


3.6288 


0.5774 


3.6800 


0.6034 


3.7316 


0.6284 


3.7844 


0.6522 


3.8374 


0.6752 


3.8914 


0.6974 


3.9456 


0.7186 


4.0004 


0.7392 


4.0558 


0.7590 


4.1114 


0.7782 


4.1678 


0.7966 


4.2244 


0.8146 


4.2814 


0.8320 


4.3388 


0.8488 


4.3966 


0.8650 


4.4546 


0.8810 


4.5132 


0.8964 


4.5718 


0.9114 


4.6308 


0.9258 


4.6900 


0.9400 


4.7496 


0.9538 


4.8094 


0.9672 


4.8692 


0.9804 


4.9294 


0.9932 


4.9898 


1.0058 


5.0504 


1.0180 



92 



T/a Q 

2.05 

2.1 

2.15 

2.2 

2.25 

2.3 

2.35 

2.4 

2.45 

2.5 

2.55 

2.6 

2.65 

2.7 

2.75 

2.8 

2.85 

2.9 

2.95 

3.0 



(B/A) 0-3 


W 0.3 /a G 


(t fi -t G )/a G 


1.6486 


5.1108 


1.0298 


1.6710 


5.1716 


1.0414 


1.6933 


5.2326 


1.0528 


1.7155 


5.2936 


1.0640 


1.7379 


5.3548 


1.0750 


1.7604 


5.4160 


1.0850 


1.7829 


5.4774 


1.0960 


1.8053 


5.5388 


1.1064 


1.8279 


5.6004 


1.1164 


1.8507 


5.6620 


1.1262 


1.8730 


5.7236 


1.1360 


1.8959 


5.7854 


1.1454 


1.9186 


5.8472 


1.1548 


1.9413 


5.9090 


1.1638 


1.9639 


5.9710 


1.1728 


1.9865 


6.0328 


1.1816 


2.0092 


6.0948 


1.1902 


2.0317 


6.1568 


1.1988 


2.0544 


6.2188 


1.2072 


2.0771 


6.2810 


1.2154 






93 



t/c g (B/A) 0<e 

0.3 1.0133 

0.35 1.0195 

0.4 1.0264 

0.45 1.0341 

0.5 1.0425 



0.55 1.0515 

0.6 1.0606 

0.65 1.0705 

0.7 1.0807 

0.75 1.0911 

0.8 1.1018 

0.85 1.1127 

0.9 1.1242 

0.95 1.1354 

1.0 1.1471 

1.05 1.1591 

1.1 1.1709 
1.15 1.1832 

1.2 1.1954 
1.25 1.2079 

1.3 1.2204 
1.35 1.2331 

1.4 1.2460 
1.45 1.2593 

1.5 1.2720 

1.55 1.2852 

1.6 1.2984 
1.65 1.3120 

1.7 1.3254 
1.75 1.3390 

1.8 1.3527 
1.85 1.3663 

1.9 1.3802 
1.95 1.3940 
2.0 1.4079 



0.50 



W 0.5 /CT G 


(t R -t Q )/a G 


2.4442 


0.2792 


2.4706 


0.3194 


2.4986 


0.3576 


2.5284 


0.3938 


2.5588 


0.4282 


2.5902 


0.4608 


2.6226 


0.4920 


2.6552 


0.5218 


2.6882 


0.5502 


2.7214 


0.5774 


2.7550 


0.6034 


2.7888 


0.6284 


2.8226 


0.6522 


2.8568 


0.6752 


2.8908 


0.6974 


2.9252 


0.7186 


2.9594 


0.7392 


2.9940 


0.7590 


3.0284 


0.7782 


3.0628 


0.7966 


3.0974 


0.8146 


3.1322 


0.8320 


3.1668 


0.8488 


3.2014 


0.8650 


3.2362 


0.8810 


3.2710 


0.8964 


3.3060 


0.9114 


3.3408 


0.9258 


3.3756 


0.9400 


3.4108 


0.9538 


3.4458 


0.9672 


3.4808 


0.9804 


3.5160 


0.9932 


3.5512 


1.0058 


3.5864 


1.0180 






94 



Va G 

2.05 

2.1 

2.15 

2.2 

2.25 

2.3 

2.35 

2.4 

2.45 

2.5 

2.55 

2.6 

2.65 

2.7 

2.75 

2.8 

2.85 

2.9 

2.95 

3.0 



(B/A) 



0.5 



1.4222 
1.4364 
1.4505 
1.4646 
1.4789 
1.4933 
1.5078 
1.5220 
1.5366 
1.5511 

1.5656 
1.5804 

1.5953 
1.6098 
1.6245 
1.6393 
1.6543 
1.6690 
1.6837 
1.6986 



W 0.5 /CT G 


(t R -t Q )/a G 


3.6216 


1.0298 


3.6570 


1.0414 


3.6924 


1.0528 


3.7280 


1.0640 


3.7634 


1.0750 


3.7988 


1.0856 


3.8344 


1.0960 


3.8702 


1.1064 


3.9056 


1.1164 


3.9414 


1.1262 


3.9772 


1.1360 


4.0130 


1.1454 


4.0486 


1.1546 


4.0844 


1.1638 


4.1204 


1.1728 


4.1564 


1.1816 


4.1922 


1.1902 


4.2282 


1.1988 


4.2644 


1.2072 


4.3002 


1.2154 



APPENDIX C-- DERIVATION OF V- . 

in j ,max 



We begin with a previously derived expression (5, p. 289) which 
relates the peak volume observed for a finite size sample, V , to the 
volume of injected sample, V" in , , and to the peak volume observed for a 
very small sample, V 

V = V p 2 +^3 (V inj ) 2 (A.1) 

Let V w = (y + 1)V p (A. 2) 

where y = loss in resolution due to a finite injection volume. 
Substituting eq A. 2 into eq A.1 and solving for V. . yields 

v inj = C3/My 2 + 2y)] 1/2 v p (a. 3) 

We now express V p in terms of N, k, and V M using eqs A.4-A.6, assuming 
V p = V b' the volurae corresponding to the base width of the peak. 

N = ( V fi/ a V ) 2 (A. 4) 

V b = ^ a y (A-5) 

V R = V M (1 + k) (A. 6) 

The result is V p = 4V M (1 + k)/N 1/2 (A. 7) 

Substitution of eq A. 7 into eq A. 3 yields the desired expression 

V inj = C3/4(y 2 + 2y] 1/2 4V M (1 + k)/N 1/2 (A. 8) 

If y is re-designated to be the maximum tolerable loss in resolution, 
then eq A. 8 becomes, upon rearrangement, 

V inj,max = 2 ^) 1/2 (y 2 + 2y) 1/2 v M (1 + k)/N 1 / 2 ( A . 9 ) 
which is the desired expression. 



95 



REFERENCES 



1. L.S. Ettre. "Evolution of Liquid Chromatography: A Historical 
Overview" in High-Performance Liquid Chromatography: Advances and 
Perspectives, Vol. 1. C. Horvath, Ed. Academic Press, New York, 
1960, pp. 2-74. 

2. H.A. Laitinen and G.W. Ewing. A History of Analytical Chemistry, 
The Maple Press Company, York, Pa., 1977, pp. 296-321. 

3. J. A. Perry. Introduction to Analytical Gas Chromatography: 
History, Principles, and Practice, Marcel Dekker, New York, 1981. 

**• L.S. Ettre. Basic Relationships of Gas Chromatography, Perkin- 
Elmer, Norwalk, Conn., 1979. 

5. L.R. Snyder and J.J. Kirkland. Introduction to Modern Liquid 
Chromatography, 2nd Ed., Wiley, New York, 1979. 

6. E. Grushka, N.M. Myers, P.D. Schetler, and J.C. Giddings. 
Computer Characterization of Chromatographic Peaks by Plate Height 
and Higher Central Moments. Anal. Chem., 41: 889-892 (1969). 

7. 0. Grubner. Interpretation of Asymmetric Curves in Linear 
Chromatography. Anal. Chem., 43: 1934-1937 (1971). 

8. L.A. Currie. Limits for Qualitative Detection and Quantitative 
Determination: Application to Radiochemistry. Anal. Chem., 40: 
586-593 (1966). 

9. H. Kaiser. Guiding Concepts in Trace Analysis. Pure Appl. Chem., 
34: 35-61 (1973). 

10. G.L. Long and J.D. Winefordner. Limit of Detection: A Closer Look 
at the TUPAC Definition. Anal. Chem. 55: 712A-724A (1983). 

11. J.C. Sternberg. "Extracolumn Contributions to Chromatographic 
Band Broadening" in Advances in Chromatography, Vol. 2. J.C. 
Giddings and R.A. Keller, Eds. Marcel Dekker, New York, 1966, pp. 
205-270. 



96 






97 



12. P.T. Kissinger, L.J. Felice, D.J. Miner, C.R. Reddy, and R.E. 
Shoup. "Detectors for Trace Organic Analysis by Liquid 
Chromatography: Principles and Applications" in Contemporary 
Topics in Analytical and Clinical Chemistry, Vol. 2. D.M. 
Hercules, G.M. Hieftje, L.R. Snyder, and M.A. Evenson, Eds. 
Plenum Press, New York, 1978, pp. 67-74, 159-175. 

13. B.L. Karger, J.N. LePage, and N. Tanaka. "Secondary Chemical 
Equilibria" in High-Performance Liquid Chromatography: Advances 
and Perspectives, Vol. 1. C. Horvath, Ed. Academic Press, New 
York, 1960, pp. 119-121. 

14. L.J. Schmauch. Response Time and Flow Sensitivity for Gas 
Chromatography. Anal. Chem., 31: 225-230 (1959). 

15. H.W. Johnson, Jr. and F.H. Stross. Gas Liquid Chromatography: 
Determination of Column Efficiency. Anal. Chem., 31: 357-364 
(1959). 

16. J.C. Giddings. Kinetic Origin of Tailing In Chromatography. 
Anal. Chem., 35: 1999-2002 (1963). 

17. R.J.E. Esser. Effect of Detector Volume on Recorded Shape of 
Chromatographic Peaks Obtained by Spectrophotometry Detection. 
Z. Anal. Chem., 236: 59-64 (1966). 

18. H.M. Gladney, B.F. Dowden, and J.D. Swalen. Computer-Assisted 
Gas-Liquid Chromatography. Anal. Chem., 41: 883-888 (1969). 

19. I.G. McWilliam and H.C. Bolton. Instrumental Peak Distortion: I. 
Relaxation Time Effects. Anal. Chem., 41: 1755-1762 (1969). 

20. I.G. McWilliam and H.G. Bolton. Instrumental Peak Distortion: II. 
Effect of Recorder Response Time. Anal. Chem., 41: 1762-1770 
(1969). 

21. V. Maynard and E. Grushka. Effect of Dead Volume on Efficiency of 
a Gas Chromatographic System. Anal. Chem., 44: 1427-1434 (1972). 

22. R.E. Pauls and L.B. Rogers. Band Broadening Studies Using 
Parameters for an Exponentially Modified Gaussian. Anal. Chem., 
49: 625-626 (1977). 

23. J.J. Kirkland, W.W. Yau, H.J. Stoklosa, and C.H. Dilks, Jr. 
Sampling and Extracolumn Effects in High Performance Liquid 
Chromatography; Influence of Peak Skew on Plate Count 
Calculations. J. Chromatogr. Sci., 15: 303-316 (1977). 

24. S.N. Chesler and S.P. Cram. Effect of Peak Sensing and Random 
Noise on the Precision and Accuracy of Statistical Moment Analyses 
from Digital Chromatographic Data. Anal. Chem., 43: 1922-1933 
(1971). 






98 

25. W.W. Yau. Characterizing Skewed Chromatographic Band Broadening. 
Anal. Chem., 49: 395-398 (1977). 

26. R.E. Pauls and L.B. Rogers. Comparison of Methods for Calculating 
Retention and Separation of Chromatographic Peaks. Sep. Sci. and 
Tech., 12: 395-413 (1977). 

27. E. Grushka. Characterization of Exponentially Modified Peaks in 
Chromatography. Anal. Chem., 44: 1733-1738 (1972). 

2a. S.M. Roberts, D.H. Wilkinson, and L.R. Walker. Practical Least 
Squares Approximation of Chromatograms. Anal. Chem., 42: 886-893 
(1970). 

29. A.H. Anderson, T.C. Gibb, and A.B. Littlewood. Computer Resolution 
of Unresolved Convolved Gas-Chromatographic Peaks. J. Chromatogr. 
Sci., 8: 640-646 (1970). 

30. I.G. McWilliam and H.C. Bolton. Instrumental Peak Distortion III. 
Analysis of Overlapping Curves. Anal. Chem., 43: 883-669 (1971). 

31. W.E. Barber and P.W. Carr. Graphical Method for Obtaining 
Retention Time and Number of Theoretical Plates from Tailed 
Chromatographic Peaks. Anal. Chem., 53: 1939-1942 (1981). 

32. P.W. Carr, University of Minnesota, Minneapolis, MN, personal 
communication, supplementary material from reference 31. 

33. M. Abramowitz and I. A. Stegun, Eds. Handbook of Mathematical 
Functions, Applied Mathematics Series No. 55. National Bureau of 
Standards, Washington, D.C. 1964, p. 932. 

34. D.G. Peters, J.M. Hayes, and G.M. Hieftje. Chemical Separations 
and Measurements, W.B. Saunders, Philadelphia, 1974, p. 18. 

35. R.S. Burington. Handbook of Mathematical Tables and Formulas, 5th 
Ed., McGraw-Hill, New York, 1973, pp. 424-427. 

36. B.L. Karger, M. Martin, and G. Guichon. Role of Column Parameters 
and Injection Volume on Detection Limits In Liquid Chromatograohy. 
Anal. Chem., 46: 1640-1647 (1974). 

37. N.H.C. Cooke, B.G. Archer, K. Olsen, and A. Berick. Comparison of 
Three- and Five-Micrometer Columns Packings for Reversed-Phase 
Liquid Chromatography. Anal. Chem., 52: 2277-2283 (1982). 

38. General Gas Chromatography Procedures. ASTM E 260-73, American 
Society for Testing & Materials, Philadelphia, Pa.; originally 
published 1965; latest revision 1973. 

39. Gas Chromatography Terms and Relationships. ASTM E 355-77, 
American Society for Testing & Materials, Philadelphia, Pa.; 
originally published 1968; latest revision 1977. 



99 



40. Liquid Chromatography Terms and Relationships. ASTM E 682-79, 
American Society for Testing & Materials, Philadelphia, Pa., 1979 ! 

Recommendations on Nomenclature for Chromatography. International 
Union of Pure and Applied Chemistry, Pure Appl. Chem., 37: 445-462 



41 



** 2 ' L " s « Ettre. The Nomenclature of Chromatography: I. Gas 
Chromatography. J. Chromatogr., 165: 235-256 (1979). 

43. L.S. Ettre. The Nomenclature of Chromatography: II. Liquid 
Chromatography. J. Chromatogr., 220: 29-63 (1981). 

44. Nomenclature, Symbols, Units and Their Usage in Spectrocheraical 
Analysis— II. Data Interpretation, International Union of Pure and 
Applied Chemistry. Spectrochimica Acta, 33B: 241-245 ( 1978) - 
rules approved 1975. 

45. Guidelines for Data Acquisition and Data Quality Evaluation in 
Environmental Chemistry. Anal. Chem., 52: 2241-2249 (1978). 

46. K. Kaiser. Quantitation In Elemental Analysis, Part II. Anal 
Chem., 42: 26A-59A (1979). 



BIOGRAPHICAL SKETCH 

Joe Preston Foley was born in Lexington, Kentucky on October 23, 
1956. He grew up on a livestock and tobacco farm outside Versailles, 
Kentucky, and attended the Woodford County public schools. His early 
adolescent interests included camping, sandlot football and basketball, 
comic books, fireworks, model rockets, and space exploration. He added 
running to his list of hobbies one day after he woke his sister (who had 
fallen asleep while sunbathing) with a firecracker. 

While attending junior high school, Joe was elected President of 
the student council and Governor of the statewide Junior Kentucky Youth 
Assembly. As he entered Woodford County High School, his extra- 
curricular interests shifted to chess and varsity tennis. In June of 
1974, he graduated with highest distinction (co-Valedictorian). 

During the next four years Joe attended Centre College of Kentucky, 
where he was inducted into Phi Beta Kappa. He received chemistry awards 
sponsored by Centre as a freshman, sophoaore, and senior. During the 
summer of 1977 he worked as a student intern for the Center for 
Technology Assessment and Policy Studies (CTAPS) at Rose-Hulraan 
Institute of Technology in Terre Haute, Indiana. In June of 1978, he 
graduated with highest distinction (co-Valedictorian) and received a 
Bachelor of Science in chemistry and chemical physics. 

In the summer of 1978 Joe again worked for the Center for Technology 
Assessment and Policy Studies. In the fall he returned to Centre 

100 



101 



College, his alma mater, for a nine-month appointment as a physical 
science laboratory instructor. 

Joe began his graduate study in chemistry at the University of 
Florida in June of 1979. In May of 1963 he was one of eleven graduate 
students selected nationally for an American Chemical Society Division 
of Analytical Chemistry Fellowship. Later in 1983 he completed the 
requirements for the degree of Doctor of Philosophy in analytical 
chemistry and accepted a National Research Council Postdoctoral Research 
Associateship at the National Bureau of Standards in Gaithersburg, 
Maryland .